- Author:
- pmr2.import <nobody@models.cellml.org>
- Date:
- 2008-07-25 04:22:31+12:00
- Desc:
- committing version01 of katsnelson_nikitina_chemla_solovyova_coirault_lecarpentier_markhasin_2004
- Permanent Source URI:
- https://models.cellml.org/workspace/katsnelson_nikitina_chemla_solovyova_coirault_lecarpentier_markhasin_2004/rawfile/5725a8a065022345930a3b20ada941f4179bb864/katsnelson_nikitina_chemla_solovyova_coirault_lecarpentier_markhasin_2004.cellml
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This CellML file was generated on 23/07/2008 at 2:08:01 at p.m. using:
COR (0.9.31.979)
Copyright 2002-2008 Dr Alan Garny
http://COR.physiol.ox.ac.uk/ - COR@physiol.ox.ac.uk
CellML 1.0 was used to generate this model
http://www.CellML.org/
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<title>Influence of viscosity on myocardium mechanical activity: a mathematical model</title>
<author>
<firstname>Geoffrey</firstname>
<surname>Nunns</surname>
<affiliation>
<shortaffil>Bioengineering Institute, The University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>
This model runs in PCEnv, but does not recreate the published results. Units are consistent throughout.
</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>
Abstract: We have previously proposed and validated a mathematical model of myocardium contraction-relaxation cycle based on current knowledge of regulatory role of Ca2+ and cross-bridge kinetics in cardiac cell. That model did not include viscous elements. Here we propose a modification of the model, in which two viscous elements are added, one in parallel to the contractile element, and one more in parallel to the series elastic element. The modified model allowed us to simulate and explain some subtle experimental data on relaxation velocity in isotonic twitches and on a mismatch between the time course of sarcomere shortening/lengthening and the time course of active force generation in isometric twitches. Model results were compared with experimental data obtained from 28 rat LV papillary muscles contracting and relaxing against various loads. Additional model analysis suggested contribution of viscosity to main inotropic and lusitropic characteristics of myocardium performance.
</para>
<para>
<informalfigure float="0" id="fig_reaction_diagram">
<mediaobject>
<imageobject>
<objectinfo>
<title>model diagram</title>
</objectinfo>
<imagedata fileref="katsnelson_2004.png"/>
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<caption>Schematic diagram of the Katsnelson et al model - an updated rheological scheme of the heart muscle including
contractile element CE, two passive elastic elements PE (parallel
one) and SE (series one) and two viscous elements VS1 and VS2. The effect of calcium and the calcium binding ligand (B) in facilitating actin-myosin binding is also highlighted.</caption>
</informalfigure>
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
<ulink url="http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WMD-4CVX3GD-2&_user=140507&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000011498&_version=1&_urlVersion=0&_userid=140507&md5=60ef8c8017472b990eae05ba7278d92f">Influence of viscosity on myocardium mechanical activity: a mathematical model</ulink>, Leonid B. Katsnelson, Larissa V. Nikitina, Denis Chemla, Olga Solovyova, Catherine Coirault, Yves Lecarpentier and Vladimir S. Markhasin, 2004, <ulink url="http://www.elsevier.com/wps/find/journaldescription.cws_home/622904/description#description">
<emphasis>Journal of Theoretical Biology</emphasis>
</ulink>, 230, 385-485. (<ulink url="http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6WMD-4CVX3GD-2-25&_cdi=6932&_user=140507&_orig=search&_coverDate=10%2F07%2F2004&_sk=997699996&view=c&wchp=dGLbVtz-zSkWb&md5=41a73e3b6fa1a75ff45f99a41d8c2a5b&ie=/sdarticle.pdf">PDF</ulink> versions of the article are available for Journal Members on the Elsevier website.) <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=15302547&dopt=Abstract">PubMed ID: 15302547 </ulink>
</para>
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<variable units="millisecond" public_interface="in" name="time"/>
<variable units="um_per_msec" public_interface="in" name="v"/>
<variable units="um_per_msec2" public_interface="in" name="phi_chi"/>
<variable units="um" public_interface="in" name="l_1"/>
<variable units="um_per_msec" public_interface="in" name="dl_1_dt"/>
<variable units="um" public_interface="in" name="l_2"/>
<variable units="um_per_msec" public_interface="in" name="dl_2_dt"/>
<variable units="um" public_interface="in" name="l_3"/>
<variable units="per_um" public_interface="in" name="alpha_1"/>
<variable units="mN" public_interface="in" name="beta_1"/>
<variable units="per_um" public_interface="in" name="alpha_2"/>
<variable units="mN" public_interface="in" name="beta_2"/>
<variable units="per_um" public_interface="in" name="alpha_3"/>
<variable units="mN" public_interface="in" name="beta_3"/>
<variable units="mN" public_interface="in" name="lambda"/>
<variable units="dimensionless" public_interface="in" name="isotonic_mode"/>
<variable units="per_um" public_interface="in" name="alpha_S"/>
<variable units="per_um" name="alpha_S_lengthening" initial_value="39"/>
<variable units="mN" name="beta_S_lengthening" initial_value="80"/>
<variable units="per_um" name="alpha_S_shortening" initial_value="46"/>
<variable units="mN" name="beta_S_shortening" initial_value="60"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci>k_S_vis</ci>
<piecewise>
<piece>
<apply>
<times/>
<ci>beta_S_lengthening</ci>
<apply>
<exp/>
<apply>
<times/>
<ci>alpha_S_lengthening</ci>
<apply>
<minus/>
<ci>l_2</ci>
<ci>l_1</ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<leq/>
<ci>dl_2_dt</ci>
<ci>dl_1_dt</ci>
</apply>
</piece>
<otherwise>
<apply>
<times/>
<ci>beta_S_shortening</ci>
<apply>
<exp/>
<apply>
<times/>
<ci>alpha_S_shortening</ci>
<apply>
<minus/>
<ci>l_2</ci>
<ci>l_1</ci>
</apply>
</apply>
</apply>
</apply>
</otherwise>
</piecewise>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci>w</ci>
</apply>
<piecewise>
<piece>
<apply>
<divide/>
<apply>
<minus/>
<apply>
<minus/>
<apply>
<times/>
<ci>k_S_vis</ci>
<apply>
<minus/>
<ci>phi_chi</ci>
<apply>
<times/>
<ci>alpha_S</ci>
<apply>
<power/>
<apply>
<minus/>
<ci>w</ci>
<ci>v</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>alpha_1</ci>
<ci>beta_1</ci>
<apply>
<exp/>
<apply>
<times/>
<ci>alpha_1</ci>
<apply>
<minus/>
<ci>l_2</ci>
<ci>l_1</ci>
</apply>
</apply>
</apply>
<apply>
<minus/>
<ci>w</ci>
<ci>v</ci>
</apply>
<cn cellml:units="um_per_msec">1</cn>
</apply>
</apply>
<apply>
<times/>
<ci>alpha_2</ci>
<ci>beta_2</ci>
<apply>
<exp/>
<apply>
<times/>
<ci>alpha_2</ci>
<ci>l_2</ci>
</apply>
</apply>
<ci>w</ci>
<cn cellml:units="um_per_msec">1</cn>
</apply>
</apply>
<ci>k_S_vis</ci>
</apply>
<apply>
<eq/>
<ci>isotonic_mode</ci>
<cn cellml:units="dimensionless">1</cn>
</apply>
</piece>
<otherwise>
<apply>
<divide/>
<apply>
<minus/>
<apply>
<minus/>
<apply>
<times/>
<ci>k_S_vis</ci>
<apply>
<minus/>
<ci>phi_chi</ci>
<apply>
<times/>
<ci>alpha_S</ci>
<apply>
<power/>
<apply>
<minus/>
<ci>w</ci>
<ci>v</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>alpha_1</ci>
<ci>beta_1</ci>
<apply>
<exp/>
<apply>
<times/>
<ci>alpha_1</ci>
<apply>
<minus/>
<ci>l_2</ci>
<ci>l_1</ci>
</apply>
</apply>
</apply>
<apply>
<minus/>
<ci>w</ci>
<ci>v</ci>
</apply>
<cn cellml:units="um_per_msec">1</cn>
</apply>
</apply>
<apply>
<times/>
<apply>
<plus/>
<apply>
<times/>
<ci>alpha_2</ci>
<ci>beta_2</ci>
<apply>
<exp/>
<apply>
<times/>
<ci>alpha_2</ci>
<ci>l_2</ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>alpha_3</ci>
<ci>beta_3</ci>
<apply>
<exp/>
<apply>
<times/>
<ci>alpha_3</ci>
<ci>l_3</ci>
</apply>
</apply>
</apply>
</apply>
<ci>w</ci>
<cn cellml:units="um_per_msec">1</cn>
</apply>
</apply>
<ci>k_S_vis</ci>
</apply>
</otherwise>
</piecewise>
</apply>
</math>
</component>
<component name="average_crossbridge_force">
<variable units="dimensionless" public_interface="out" name="p_v"/>
<variable units="per_millisecond" public_interface="out" name="p_prime_v"/>
<variable units="dimensionless" public_interface="out" name="P_star"/>
<variable units="dimensionless" public_interface="out" name="G_star"/>
<variable units="dimensionless" name="gamma"/>
<variable units="per_millisecond" name="case_1"/>
<variable units="per_millisecond" name="case_2"/>
<variable units="per_millisecond" name="case_3"/>
<variable units="per_millisecond" name="case_4"/>
<variable units="um_per_msec" public_interface="in" name="v"/>
<variable units="um_per_msec2" public_interface="in" name="phi_chi"/>
<variable units="um_per_msec" public_interface="in" name="v_1"/>
<variable units="um_per_msec" public_interface="in" name="v_max"/>
<variable units="dimensionless" public_interface="in" name="a"/>
<variable units="dimensionless" public_interface="in" name="d_h"/>
<variable units="dimensionless" public_interface="in" name="alpha_G"/>
<variable units="dimensionless" public_interface="in" name="alpha_P"/>
<variable units="dimensionless" public_interface="in" name="alpha_PG"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<ci>gamma</ci>
<apply>
<divide/>
<apply>
<times/>
<ci>a</ci>
<ci>d_h</ci>
<apply>
<power/>
<apply>
<divide/>
<ci>v_1</ci>
<ci>v_max</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
<apply>
<minus/>
<apply>
<times/>
<cn cellml:units="dimensionless">3</cn>
<ci>a</ci>
<ci>d_h</ci>
</apply>
<apply>
<divide/>
<apply>
<times/>
<apply>
<plus/>
<ci>a</ci>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>v</ci>
</apply>
<ci>v_max</ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>P_star</ci>
<piecewise>
<piece>
<apply>
<divide/>
<apply>
<times/>
<ci>a</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">1</cn>
<apply>
<divide/>
<ci>v</ci>
<ci>v_max</ci>
</apply>
</apply>
</apply>
<apply>
<minus/>
<ci>a</ci>
<apply>
<divide/>
<ci>v</ci>
<ci>v_max</ci>
</apply>
</apply>
</apply>
<apply>
<leq/>
<ci>v</ci>
<cn cellml:units="um_per_msec">0</cn>
</apply>
</piece>
<otherwise>
<apply>
<minus/>
<apply>
<plus/>
<cn cellml:units="dimensionless">1</cn>
<ci>d_h</ci>
</apply>
<apply>
<divide/>
<apply>
<times/>
<apply>
<power/>
<ci>d_h</ci>
<cn cellml:units="dimensionless">2</cn>
</apply>
<ci>a</ci>
</apply>
<apply>
<plus/>
<apply>
<times/>
<apply>
<divide/>
<apply>
<times/>
<ci>a</ci>
<ci>d_h</ci>
</apply>
<ci>gamma</ci>
</apply>
<apply>
<power/>
<apply>
<divide/>
<ci>v</ci>
<ci>v_max</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
<apply>
<divide/>
<apply>
<times/>
<apply>
<plus/>
<ci>a</ci>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>v</ci>
</apply>
<ci>v_max</ci>
</apply>
<apply>
<times/>
<ci>a</ci>
<ci>d_h</ci>
</apply>
</apply>
</apply>
</apply>
</otherwise>
</piecewise>
</apply>
<apply>
<eq/>
<ci>G_star</ci>
<piecewise>
<piece>
<apply>
<plus/>
<cn cellml:units="dimensionless">1</cn>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="dimensionless">0.6</cn>
<ci>v</ci>
</apply>
<ci>v_max</ci>
</apply>
</apply>
<apply>
<and/>
<apply>
<leq/>
<apply>
<minus/>
<ci>v_max</ci>
</apply>
<ci>v</ci>
</apply>
<apply>
<leq/>
<ci>v</ci>
<cn cellml:units="um_per_msec">0</cn>
</apply>
</apply>
</piece>
<piece>
<apply>
<divide/>
<ci>P_star</ci>
<apply>
<plus/>
<apply>
<divide/>
<apply>
<times/>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="dimensionless">0.4</cn>
<ci>a</ci>
</apply>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>a</ci>
</apply>
<ci>v</ci>
</apply>
<ci>v_max</ci>
</apply>
<cn cellml:units="dimensionless">1</cn>
</apply>
</apply>
<apply>
<and/>
<apply>
<lt/>
<cn cellml:units="um_per_msec">0</cn>
<ci>v</ci>
</apply>
<apply>
<leq/>
<ci>v</ci>
<ci>v_1</ci>
</apply>
</apply>
</piece>
<otherwise>
<apply>
<divide/>
<apply>
<times/>
<ci>P_star</ci>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<ci>alpha_G</ci>
</apply>
<apply>
<power/>
<apply>
<divide/>
<apply>
<minus/>
<ci>v</ci>
<ci>v_1</ci>
</apply>
<ci>v_max</ci>
</apply>
<ci>alpha_PG</ci>
</apply>
</apply>
</apply>
</apply>
<apply>
<plus/>
<apply>
<divide/>
<apply>
<times/>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="dimensionless">0.4</cn>
<ci>a</ci>
</apply>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>a</ci>
</apply>
<ci>v</ci>
</apply>
<ci>v_max</ci>
</apply>
<cn cellml:units="dimensionless">1</cn>
</apply>
</apply>
</otherwise>
</piecewise>
</apply>
<apply>
<eq/>
<ci>case_1</ci>
<apply>
<divide/>
<apply>
<times/>
<ci>a</ci>
<apply>
<plus/>
<cn cellml:units="um_per_msec2">0.4</cn>
<apply>
<times/>
<cn cellml:units="um_per_msec2">0.4</cn>
<ci>a</ci>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>v_max</ci>
<apply>
<power/>
<apply>
<times/>
<apply>
<plus/>
<ci>a</ci>
<cn cellml:units="dimensionless">1</cn>
</apply>
<cn cellml:units="dimensionless">0.4</cn>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>case_2</ci>
<apply>
<divide/>
<apply>
<times/>
<ci>a</ci>
<cn cellml:units="um_per_msec2">1</cn>
<apply>
<plus/>
<cn cellml:units="dimensionless">1</cn>
<apply>
<times/>
<cn cellml:units="dimensionless">0.4</cn>
<ci>a</ci>
</apply>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="dimensionless">1.2</cn>
<ci>v</ci>
</apply>
<ci>v_max</ci>
</apply>
<apply>
<times/>
<cn cellml:units="dimensionless">0.6</cn>
<apply>
<power/>
<apply>
<divide/>
<ci>v</ci>
<ci>v_max</ci>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>v_max</ci>
<apply>
<power/>
<apply>
<times/>
<apply>
<minus/>
<ci>a</ci>
<apply>
<divide/>
<ci>v</ci>
<ci>v_max</ci>
</apply>
</apply>
<apply>
<plus/>
<cn cellml:units="dimensionless">1</cn>
<apply>
<divide/>
<apply>
<times/>
<cn cellml:units="dimensionless">0.6</cn>
<ci>v</ci>
</apply>
<ci>v_max</ci>
</apply>
</apply>
</apply>
<cn cellml:units="dimensionless">2</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>case_3</ci>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="um_per_msec2">0.4</cn>
<ci>a</ci>
</apply>
<cn cellml:units="um_per_msec2">1</cn>
</apply>
<apply>
<times/>
<ci>a</ci>
<ci>v_max</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>case_4</ci>
<apply>
<times/>
<apply>
<divide/>
<cn cellml:units="um_per_msec2">1</cn>
<ci>v_max</ci>
</apply>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<ci>alpha_G</ci>
</apply>
<apply>
<power/>
<apply>
<minus/>
<apply>
<divide/>
<ci>v</ci>
<ci>v_max</ci>
</apply>
<apply>
<divide/>
<ci>v_1</ci>
<ci>v_max</ci>
</apply>
</apply>
<ci>alpha_PG</ci>
</apply>
</apply>
</apply>
<apply>
<plus/>
<apply>
<divide/>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="dimensionless">0.4</cn>
<ci>a</ci>
</apply>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>a</ci>
</apply>
<apply>
<times/>
<ci>alpha_G</ci>
<ci>alpha_P</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">1</cn>
<apply>
<divide/>
<apply>
<times/>
<apply>
<plus/>
<apply>
<times/>
<cn cellml:units="dimensionless">0.4</cn>
<ci>a</ci>
</apply>
<cn cellml:units="dimensionless">1</cn>
</apply>
<ci>v</ci>
</apply>
<apply>
<times/>
<ci>a</ci>
<ci>v_max</ci>
</apply>
</apply>
</apply>
</apply>
</apply>
<apply>
<power/>
<apply>
<minus/>
<apply>
<divide/>
<ci>v</ci>
<ci>v_max</ci>
</apply>
<apply>
<divide/>
<ci>v_1</ci>
<ci>v_max</ci>
</apply>
</apply>
<apply>
<minus/>
<ci>alpha_PG</ci>
<cn cellml:units="dimensionless">1</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>p_v</ci>
<apply>
<divide/>
<ci>P_star</ci>
<ci>G_star</ci>
</apply>
</apply>
<apply>
<eq/>
<ci>p_prime_v</ci>
<piecewise>
<piece>
<apply>
<times/>
<ci>case_1</ci>
<ci>phi_chi</ci>
</apply>
<apply>
<leq/>
<ci>v</ci>
<apply>
<minus/>
<ci>v_max</ci>
</apply>
</apply>
</piece>
<piece>
<apply>
<times/>
<ci>case_2</ci>
<ci>phi_chi</ci>
</apply>
<apply>
<and/>
<apply>
<lt/>
<apply>
<minus/>
<ci>v_max</ci>
</apply>
<ci>v</ci>
</apply>
<apply>
<leq/>
<ci>v</ci>
<cn cellml:units="um_per_msec">0</cn>
</apply>
</apply>
</piece>
<piece>
<apply>
<times/>
<ci>case_3</ci>
<ci>phi_chi</ci>
</apply>
<apply>
<and/>
<apply>
<lt/>
<cn cellml:units="um_per_msec">0</cn>
<ci>v</ci>
</apply>
<apply>
<leq/>
<ci>v</ci>
<ci>v_1</ci>
</apply>
</apply>
</piece>
<otherwise>
<apply>
<times/>
<ci>case_4</ci>
<ci>phi_chi</ci>
</apply>
</otherwise>
</piecewise>
</apply>
</math>
</component>
<component name="calcium_handling">
<variable units="dimensionless" public_interface="out" name="A" initial_value="0"/>
<variable units="per_millisecond" name="dA_dt"/>
<variable units="dimensionless" name="N_A"/>
<variable units="dimensionless" name="pi_N_A"/>
<variable units="dimensionless" public_interface="out" name="B" initial_value="0"/>
<variable units="per_millisecond" name="dB_dt"/>
<variable units="dimensionless" public_interface="out" name="Ca_C" initial_value="0"/>
<variable units="millisecond" public_interface="in" name="time"/>
<variable units="dimensionless" public_interface="in" name="N"/>
<variable units="per_millisecond" public_interface="in" name="a_on"/>
<variable units="per_millisecond" public_interface="in" name="a_off"/>
<variable units="dimensionless" public_interface="in" name="k_A"/>
<variable units="dimensionless" public_interface="in" name="L_oz"/>
<variable units="dimensionless" name="A_tot" initial_value="1"/>
<variable units="dimensionless" name="B_tot" initial_value="0.4"/>
<variable units="per_millisecond" name="b_on" initial_value="2.6"/>
<variable units="per_millisecond" name="b_off" initial_value="0.182"/>
<variable units="per_millisecond2" name="a_c" initial_value="0.005"/>
<variable units="per_millisecond2" name="b_c" initial_value="0.03"/>
<variable units="per_millisecond" name="r_Ca" initial_value="0.65"/>
<variable units="dimensionless" name="q_Ca" initial_value="50"/>
<variable units="millisecond" name="t_d" initial_value="33"/>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci>A</ci>
</apply>
<ci>dA_dt</ci>
</apply>
<apply>
<eq/>
<ci>dA_dt</ci>
<apply>
<minus/>
<apply>
<times/>
<ci>a_on</ci>
<apply>
<minus/>
<ci>A_tot</ci>
<ci>A</ci>
</apply>
<ci>Ca_C</ci>
</apply>
<apply>
<times/>
<ci>a_off</ci>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
<ci>k_A</ci>
</apply>
<ci>A</ci>
</apply>
</apply>
<ci>pi_N_A</ci>
<ci>A</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>N_A</ci>
<apply>
<divide/>
<ci>N</ci>
<apply>
<times/>
<ci>L_oz</ci>
<ci>A</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<ci>pi_N_A</ci>
<piecewise>
<piece>
<cn cellml:units="dimensionless">1</cn>
<apply>
<geq/>
<ci>N_A</ci>
<cn cellml:units="dimensionless">1</cn>
</apply>
</piece>
<otherwise>
<apply>
<power/>
<cn cellml:units="dimensionless">0.02</cn>
<ci>N_A</ci>
</apply>
</otherwise>
</piecewise>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci>B</ci>
</apply>
<ci>dB_dt</ci>
</apply>
<apply>
<eq/>
<ci>dB_dt</ci>
<apply>
<minus/>
<apply>
<times/>
<ci>b_on</ci>
<apply>
<minus/>
<ci>B_tot</ci>
<ci>B</ci>
</apply>
<ci>Ca_C</ci>
</apply>
<apply>
<times/>
<ci>b_off</ci>
<ci>B</ci>
</apply>
</apply>
</apply>
<apply>
<eq/>
<apply>
<diff/>
<bvar>
<ci>time</ci>
</bvar>
<ci>Ca_C</ci>
</apply>
<piecewise>
<piece>
<apply>
<times/>
<ci>b_c</ci>
<ci>time</ci>
<apply>
<minus/>
<cn cellml:units="dimensionless">1</cn>
<apply>
<exp/>
<apply>
<times/>
<apply>
<minus/>
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