Location: Fenton, Karma, 1998 @ 40a1d95f3797 / fenton_karma_1998.cellml

Author:

pmr2.import <nobody@models.cellml.org>

Date:

2006-07-09 07:19:47+12:00

Desc:

committing version01 of fenton_karma_1998

Permanent Source URI:

https://models.cellml.org/workspace/5bb/rawfile/40a1d95f37972af3c904c777e04b97570b3f99a6/fenton_karma_1998.cellml

<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : fenton_karma_model_1998.xml

CREATED : 24th June 2003

LAST MODIFIED : 21st April 2005

AUTHOR : Catherine Lloyd
         Bioengineering Institute
         The University of Auckland
         
MODEL STATUS : This model conforms to the CellML 1.0 Specification released on
10th August 2001, and the 16/01/2002 CellML Metadata 1.0 Specification.

DESCRIPTION : This file contains a CellML description of the Fenton-Karma
simplifid ionic model with three membrane currents, 1998. 

CHANGES: 
29/07/2003 - CML - Altered some parameter values in accordance with some
published errata.  
21/04/2005 - PJV - Changed units to correct inconsistant dimensions. 
--><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="fenton_karma_1998_version01" name="fenton_karma_1998_version01">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
  <articleinfo>
  <title>A Simplified Ventricular Myocyte Model</title>
  <author>
    <firstname>Catherine</firstname>
          <surname>Lloyd</surname>
    <affiliation>
      <shortaffil>Bioengineering Institute, University of Auckland</shortaffil>
    </affiliation>
  </author>
</articleinfo>
  <section id="sec_status">
    <title>Model Status</title>
    <para>
            This is the original unchecked version of the model imported from the previous
            CellML model repository, 24-Jan-2006.
          </para>
  </section>
  <sect1 id="sec_structure">
<title>Model Structure</title>

<para>
Ventricular fibrillation (VF) is a disorganised electrical wave activity that disrupts the regular and synchronised contraction of the ventricular muscle and consequently destroys the main pumping function of the heart.  Research to date has suggested that electrical vortices are the mechanism underlying VF.  These are manisfested in the 2D heart as spiral waves, and in the 3D heart as scroll waves of action potential.  Compared with the extensive knowledge available on vortex filament behaviour in an isotropic medium, relatively little is known about how the vortices behave in the anisotropic ventricular muscle.  This 1998 publication by Fenton and Karma (fully referenced below) explores the dynamics of the vortex filament in continuous myocardium via numerical simulation.    </para>

<para>
Over the past 25 to 30 years, mathematical models that describe ventricular action potential have become increasingly complex as new experimental data has become available and has been incorporated into the mathematical equations.  Although these complex models are more realistic, they are also computationally expensive to run, and isolating subsets of essential parameters from the model is difficult.  One traditional method for avoiding this complexity is to use simplified models such as <ulink url="${HTML_EXMPL_FN_SIMPLE}">the FitzHugh-Nagumo model, 1961</ulink>.  However, these simplified models have been criticised as being too simple and as not having the capacity to fully capture certain important features of the ventricular action potential.
</para>  

<para>
The modelling approach that Fenton and Karma take is to use a simplified ionic model of ventricular action potential with three membrane currents.  This model retains enough detail to quantitatively reproduce the behaviour of the ventricular action potential captured by the more complex ionic models of cardiac action potential (such as <ulink url="${HTML_EXMPL_BR_MODEL}">the Beeler-Reuter 1977 model</ulink>, and <ulink url="${HTML_EXMPL_LR_I_MODEL}">the original Luo-Rudy 1991 model</ulink>), but it is less computationally expensive than these other models.
</para>

<para>
The three currents in the Fenton-Karma model are: 
</para>
<itemizedlist>
  <listitem>
            <para>
              <emphasis>I<subscript>fi</subscript>
              </emphasis>, a fast inward current which corresponds to the <emphasis>I<subscript>Na</subscript>
              </emphasis> current</para>
          </listitem>;
  <listitem>
            <para>
              <emphasis>I<subscript>so</subscript>
              </emphasis>, a slow outward current which corresponds to the <emphasis>I<subscript>K</subscript>
              </emphasis> current</para>
          </listitem>; and     
  <listitem>
            <para>
              <emphasis>I<subscript>si</subscript>
              </emphasis>, a slow inward current which corresponds to the <emphasis>I<subscript>Ca</subscript>
              </emphasis> current</para>
          </listitem>.
</itemizedlist> 
<para>
(see <xref linkend="fig_cell_diagram"/> below).
The authors choose to use these labels as opposed to Na, K and Ca, as a reminder that <emphasis>I<subscript>fi</subscript>
          </emphasis>, <emphasis>I<subscript>so</subscript>
          </emphasis>, and  <emphasis>I<subscript>si</subscript>
          </emphasis> do not actually represent measured currents, but only their activation, inactivation and reactivation dynamics which are needed to quantitatively reproduce restitution properties.
</para>

<para>
Model validation indicates that this simplified model is able to faithfully reproduce the 2D patterns of reentry of the more complex models.
</para>

<para>
The complete original paper reference is cited below:
</para>

<para>
<ulink url="http://ojps.aip.org/getabs/servlet/GetabsServlet?prog=normal&amp;id=CHAOEH000008000001000020000001&amp;idtype=cvips&amp;gifs=Yes">Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation</ulink>, Flavio Fenton and Alain Karma, 1998, <ulink url="http://ojps.aip.org/journals/doc/CHAOEH-home/top.jsp">
            <emphasis>Chaos</emphasis>
          </ulink>, 8, 20-47.  (Full text (HTML) and PDF versions of the article are available to subscribers on the <emphasis>Chaos</emphasis> website.)  <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&amp;db=PubMed&amp;list_uids=12779708&amp;dopt=Abstract">PubMed ID: 12779708</ulink>
</para>

<informalfigure float="0" id="fig_cell_diagram">
<mediaobject>
  <imageobject>
    <objectinfo>
      <title>cell diagram</title>
    </objectinfo>
    <imagedata fileref="fenton_1998a.png"/>
  </imageobject>
</mediaobject>
<caption>A schematic diagram of the three ionic currents described by the Fenton-Karma model of a ventricular myocyte.</caption>
</informalfigure>

<para>
The Fenton-Karma model has been described here in CellML (the raw CellML description of the Fenton-Karma model can be downloaded in various formats as described in <xref linkend="sec_download_this_model"/>).  The parameter values used in this version of the Fenton-Karma model are consistent with the modified Beeler-Reuter (see Table 1 of the 1998 model errata).  Simulations of this CellML model can be run using CMISS.        
</para>

</sect1>
</article>
</documentation>
  
  
  <!--
    Below, we define some additional units for association with variables and
    constants within the model. 
  -->
  
  <units name="ms">
    <unit units="second" prefix="milli"/>
  </units>
  
  <units name="per_ms">
    <unit units="second" prefix="milli" exponent="-1"/>
  </units>

  <units name="mV">
    <unit units="volt" prefix="milli"/>
  </units>

  <units name="per_mV">
    <unit units="volt" prefix="milli" exponent="-1"/>
  </units>
  
  <units name="per_mV_ms">
    <unit units="mV" exponent="-1"/>
    <unit units="ms" exponent="-1"/>
  </units>
  
  <units name="mS_per_mm2">
    <unit units="siemens" prefix="milli"/>
    <unit units="metre" prefix="milli" exponent="-2"/>
  </units>

  <units name="uF_per_mm2">
    <unit units="farad" prefix="micro"/>
    <unit units="metre" prefix="milli" exponent="-2"/>
  </units>

  <units name="uA_per_mm2">
    <unit units="ampere" prefix="micro"/>
    <unit units="metre" prefix="milli" exponent="-2"/>
  </units>
  
  <units name="concentration_units">
    <unit units="mole" prefix="nano"/>
    <unit units="metre" prefix="milli" exponent="-3"/>
  </units>
    
  <!--
    The "environment" component is used to declare variables that are used by
    all or most of the other components, in this case just "time".
  -->
  <component name="environment">
    <variable units="ms" public_interface="out" name="time"/>
  </component>
  
  <component name="membrane">
    <variable units="dimensionless" public_interface="out" name="u" initial_value="0.0"/>
    
    <variable units="uF_per_mm2" public_interface="out" name="Cm" initial_value="0.01"/>
    <variable units="mV" name="Vm"/>
    <variable units="mV" name="V_o" initial_value="-85.0"/>
    <variable units="mV" name="V_fi" initial_value="15.0"/>
    <variable units="uA_per_mm2" name="I_stim"/>
    
    <variable units="ms" public_interface="in" name="time"/>
    <variable units="per_ms" public_interface="in" name="J_fi"/>
    <variable units="per_ms" public_interface="in" name="J_so"/>
    <variable units="per_ms" public_interface="in" name="J_si"/>
    <variable units="per_ms" name="J_stim"/>
    <variable units="per_ms" name="total_flux"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="total_flux_calculation">
        <eq/>
        <ci>total_flux</ci>
        <apply>
          <minus/>
          <ci>J_stim</ci>
          <apply>
            <plus/>
            <ci> J_fi </ci>
            <ci> J_so </ci>
            <ci> J_si </ci>
          </apply>
        </apply>
      </apply>
      <apply id="membrane_voltage_diff_eq">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> u </ci>
        </apply>
        <ci>total_flux</ci>
      </apply>
    </math>
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="J_stim_calculation">
        <eq/>
        <ci>J_stim</ci>
        <apply>
          <divide/>
          <ci>I_stim</ci>
          <apply>
            <times/>
            <ci>Cm</ci>
            <apply>
              <minus/>
              <ci>V_fi</ci>
              <ci>V_o</ci>
            </apply>
          </apply>
        </apply>
      </apply>
    </math>
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="Vm_calculation">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> Vm </ci>
        </apply>
        <apply>
          <times/>
          <ci>total_flux</ci>
          <apply>
            <minus/>
            <ci>V_fi</ci>
            <ci>V_o</ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="p">
    <variable units="dimensionless" public_interface="out" name="p"/>
    <variable units="mV" public_interface="out" name="u_c" initial_value="0.13"/>
    
    <variable units="dimensionless" public_interface="in" name="u"/>
    <variable units="ms" public_interface="in" name="time"/>
  
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="p_calculation">
        <eq/>
        <ci> p </ci>
        <piecewise>
          <piece>
            <cn cellml:units="dimensionless"> 0.0 </cn>
            <apply>
              <lt/>
              <ci> u </ci> 
              <ci> u_c </ci>
            </apply>
          </piece>
          <otherwise>
            <cn cellml:units="dimensionless"> 1.0 </cn>
          </otherwise>
        </piecewise>
      </apply>
    </math>
  </component> 
  
  <component name="q">
    <variable units="dimensionless" public_interface="out" name="q"/>
    
    <variable units="dimensionless" name="u_v" initial_value="0.055"/>
    
    <variable units="dimensionless" public_interface="in" name="u"/>
    <variable units="ms" public_interface="in" name="time"/>
  
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="q_calculation">
        <eq/>
        <ci> q </ci>
        <piecewise>
          <piece>
            <cn cellml:units="dimensionless"> 0.0 </cn>
            <apply>
              <lt/>
              <ci> u </ci> 
              <ci> u_v </ci>
            </apply>
          </piece>
          <otherwise>
            <cn cellml:units="dimensionless"> 1.0 </cn>
          </otherwise>
        </piecewise>
      </apply>
    </math>
  </component> 
  
  <component name="fast_inward_current">
    <variable units="per_ms" public_interface="out" name="J_fi"/>
  
    <variable units="ms" name="tau_d"/>
    <variable units="mS_per_mm2" name="g_fi_max" initial_value="0.04"/>
    <variable units="uF_per_mm2" public_interface="in" name="Cm"/>
    
    <variable units="dimensionless" public_interface="in" private_interface="out" name="p"/>
    <variable units="dimensionless" public_interface="in" private_interface="out" name="q"/>
    <variable units="mV" public_interface="in" name="u_c"/>
    <variable units="ms" public_interface="in" private_interface="out" name="time"/>
    <variable units="dimensionless" public_interface="in" private_interface="out" name="u"/>
    
    <variable units="dimensionless" private_interface="in" name="v"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="tau_d_calculation">
        <eq/>
        <ci>tau_d</ci>
        <apply>
          <divide/>
          <ci>Cm</ci>
          <ci>g_fi_max</ci>
        </apply>
      </apply>
      
      <apply id="J_fi_calculation">
        <eq/>
        <ci> J_fi </ci>
        <apply>
          <times/>
          <apply>
            <minus/>
            <ci> v </ci>
          </apply>
          <ci> p </ci>
          <apply>
            <minus/>
            <ci> u </ci>
            <ci> u_c </ci>
          </apply>
          <apply>
            <divide/>
            <apply>
              <minus/>
              <cn cellml:units="dimensionless"> 1.0 </cn>
              <ci> u </ci>
            </apply>
            <ci> tau_d </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="fast_inward_current_v_gate">
    <variable units="dimensionless" public_interface="out" name="v" initial_value="1.0"/>
    
    <variable units="ms" name="tau_v_minus"/>
    <variable units="ms" name="tau_v1_minus" initial_value="1000.0"/>
    <variable units="ms" name="tau_v2_minus" initial_value="19.2"/>
    <variable units="ms" name="tau_v_plus" initial_value="3.33"/>
    
    <variable units="dimensionless" public_interface="in" name="q"/>
    <variable units="dimensionless" public_interface="in" name="p"/>
    <variable units="dimensionless" public_interface="in" name="u"/>
    <variable units="ms" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="dv_dt">
        <eq/>
        <apply>
          <diff/>
          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> v </ci>
        </apply>
        <apply>
          <minus/>
          <apply>
            <times/>
            <apply>
              <minus/>
              <cn cellml:units="dimensionless"> 1.0 </cn>
              <ci> p </ci>
            </apply>
            <apply>
              <divide/>
              <apply>
                <minus/>
                <cn cellml:units="dimensionless"> 1.0 </cn>
                <ci> v </ci>
              </apply>
              <ci> tau_v_minus </ci>
            </apply>
          </apply>
          <apply>
            <divide/>
            <apply>
              <times/>
              <ci> p </ci>
              <ci> v </ci>
            </apply>
            <ci> tau_v_plus </ci>
          </apply>
        </apply>
      </apply>
      
      <apply id="tau_v_minus_calculation">
        <eq/>
        <ci> tau_v_minus </ci>
        <apply>
          <plus/>
          <apply>
            <times/>
            <ci> q </ci>
            <ci> tau_v1_minus </ci>
          </apply>
          <apply>
            <times/>
            <apply>
              <minus/>
              <cn cellml:units="dimensionless"> 1.0 </cn>
              <ci> q </ci>
            </apply>
            <ci> tau_v2_minus </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="slow_outward_current">
    <variable units="per_ms" public_interface="out" name="J_so"/>
  
    <variable units="ms" name="tau_0" initial_value="8.3"/>
    <variable units="ms" name="tau_r" initial_value="50.0"/>
    
    <variable units="dimensionless" public_interface="in" name="p"/>
    <variable units="ms" public_interface="in" name="time"/>
    <variable units="dimensionless" public_interface="in" name="u"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="J_so_calculation">
        <eq/>
        <ci> J_so </ci>
        <apply>
          <plus/>
          <apply>
            <divide/>
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              <times/>
              <ci> u </ci>
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                <minus/>
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            </apply>
            <ci> tau_0 </ci>
          </apply>
          <apply>
            <divide/>
            <ci> p </ci>
            <ci> tau_r </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component> 
  
  <component name="slow_inward_current">
    <variable units="per_ms" public_interface="out" name="J_si"/>
  
    <variable units="ms" name="tau_si" initial_value="44.84"/>
    <variable units="dimensionless" name="u_csi" initial_value="0.85"/>
    <variable units="dimensionless" name="k" initial_value="10.0"/>
    
    <variable units="dimensionless" public_interface="in" private_interface="out" name="p"/>
    <variable units="ms" public_interface="in" private_interface="out" name="time"/>
    <variable units="dimensionless" public_interface="in" private_interface="out" name="u"/>
    
    <variable units="dimensionless" private_interface="in" name="w"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="J_si_calculation">
        <eq/>
        <ci> J_si </ci>
        <apply>
          <divide/>
          <apply>
            <times/>
            <apply>
              <minus/>
              <ci> w </ci>
            </apply>
            <apply>
              <plus/>
              <cn cellml:units="dimensionless"> 1.0 </cn>
              <apply>
                <tanh/>
                <apply>
                  <times/>
                  <ci> k </ci>
                  <apply>
                    <minus/>
                    <ci> u </ci>
                    <ci> u_csi </ci>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
          <apply>
            <times/>
            <cn cellml:units="dimensionless"> 2.0 </cn>
            <ci> tau_si </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <component name="slow_inward_current_w_gate">
    <variable units="dimensionless" public_interface="out" name="w" initial_value="1.0"/>
    
    <variable units="ms" name="tau_w_minus" initial_value="11.0"/>
    <variable units="ms" name="tau_w_plus" initial_value="667.0"/>
    
    <variable units="dimensionless" public_interface="in" name="p"/>
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    <variable units="ms" public_interface="in" name="time"/>
    
    <math xmlns="http://www.w3.org/1998/Math/MathML">
      <apply id="dw_dt">
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          <bvar>
            <ci> time </ci>
          </bvar>
          <ci> w </ci>
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        <apply>
          <minus/>
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            <times/>
            <apply>
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              <ci> p </ci>
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              <divide/>
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                <ci> w </ci>
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              <ci> tau_w_minus </ci>
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            <divide/>
            <apply>
              <times/>
              <ci> p </ci>
              <ci> w </ci>
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            <ci> tau_w_plus </ci>
          </apply>
        </apply>
      </apply>
    </math>
  </component>
  
  <group>
    <relationship_ref relationship="containment"/>
    <component_ref component="membrane">
      <component_ref component="fast_inward_current">
        <component_ref component="fast_inward_current_v_gate"/>
      </component_ref>
      <component_ref component="slow_outward_current"/>
      <component_ref component="slow_inward_current">
        <component_ref component="slow_inward_current_w_gate"/>
      </component_ref>
      <component_ref component="p"/>
      <component_ref component="q"/>
    </component_ref>
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  <group>
    <relationship_ref relationship="encapsulation"/>
    <component_ref component="fast_inward_current">
      <component_ref component="fast_inward_current_v_gate"/>
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    <component_ref component="slow_inward_current">
      <component_ref component="slow_inward_current_w_gate"/>
    </component_ref>
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  <connection>
    <map_components component_2="environment" component_1="membrane"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>

  <connection>
    <map_components component_2="environment" component_1="p"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="environment" component_1="q"/>
    <map_variables variable_2="time" variable_1="time"/>
  </connection>
  
  <connection>
    <map_components component_2="environment" component_1="fast_inward_current"/>
    <map_variables variable_2="time" variable_1="time"/>
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    <map_components component_2="environment" component_1="slow_outward_current"/>
    <map_variables variable_2="time" variable_1="time"/>
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    <map_components component_2="fast_inward_current" component_1="membrane"/>
    <map_variables variable_2="u" variable_1="u"/>
    <map_variables variable_2="J_fi" variable_1="J_fi"/>
    <map_variables variable_2="Cm" variable_1="Cm"/>
  </connection>  
  
  <connection>
    <map_components component_2="slow_outward_current" component_1="membrane"/>
    <map_variables variable_2="u" variable_1="u"/>
    <map_variables variable_2="J_so" variable_1="J_so"/>
  </connection>
  
  <connection>
    <map_components component_2="slow_inward_current" component_1="membrane"/>
    <map_variables variable_2="u" variable_1="u"/>
    <map_variables variable_2="J_si" variable_1="J_si"/>
  </connection>
  
  <connection>
    <map_components component_2="p" component_1="membrane"/>
    <map_variables variable_2="u" variable_1="u"/>
  </connection>
  
  <connection>
    <map_components component_2="q" component_1="membrane"/>
    <map_variables variable_2="u" variable_1="u"/>
  </connection>
  
  <connection>
    <map_components component_2="fast_inward_current" component_1="p"/>
    <map_variables variable_2="p" variable_1="p"/>
    <map_variables variable_2="u_c" variable_1="u_c"/>
  </connection>
  
  <connection>
    <map_components component_2="slow_outward_current" component_1="p"/>
    <map_variables variable_2="p" variable_1="p"/>
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  <connection>
    <map_components component_2="slow_inward_current" component_1="p"/>
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  <connection>
    <map_components component_2="fast_inward_current" component_1="q"/>
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  <connection>
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          Altered some parameter values in accordance with some published 
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          This is the CellML description of the Fenton-Karma simplified ionic 
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          The Fenton Karma model is based on a non-dimensional "potential" 
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        The University of Auckland, Bioengineering Institute
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          Changed units to correct inconsistant dimensions
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        The Fenton-Karma simplified ionic model with three membrane currents, 
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            Vortex dynamics in three-dimensional continuous myocardium with 
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