- Author:
- pmr2.import <nobody@models.cellml.org>
- Date:
- 2009-06-17 14:28:53+12:00
- Desc:
- committing version03 of guccione_mcculloch_waldman_1991
- Permanent Source URI:
- https://models.cellml.org/workspace/guccione_mcculloch_waldman_1991/rawfile/c6aa63b915f9dfdf3ed4e7a99f9bcc6a89bf6980/guccione_mcculloch_waldman_1991.cellml
<?xml version='1.0' encoding='utf-8'?>
<!-- FILE : pole_zero_law_1991.xml
CREATED : 30th July 2003
LAST MODIFIED : 30th July 2003
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 16/1/02 CellML Metadata 1.0 Specification.
DESCRIPTION : This file contains a CellML description of the Pole Zero constitutive material law, defining the relation between the six independent strain components and the stress components.
CHANGES:
18/02/04 - CML - Completed the Metadata.
--><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="PoleZero" name="guccione_mcculloch_waldman_1991_version03">
<documentation xmlns="http://cellml.org/tmp-documentation">
<article>
<articleinfo>
<title>The Guccione Constitutive Material Law</title>
<author>
<firstname>Vijayaraghavan</firstname>
<surname>Rajagopal</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>
This is an example of how CellML can be used to describe a material law which models the passive, mechanical behaviour of a material. The material law is to calculate the stress induced in a body when a certain amount of deformation (strain) is imposed on it. The calculated stresses are used to solve a system of equations that govern the mechanics of large deformations. This is an example of how CellML can be used to describe a material law. The example presented is the Guccione constitutive material law, which defines the relationship between six independent strain components and the stress components.
</para>
<para>
The Guccione material law was developed by Guccione <emphasis>et al.</emphasis> to model the passive material (mechanical) properties of intact ventricular myocardium.They approximated the equatorial region of the canine left ventricle as a thich-walled cylinder made of incompressible (volume is conserved during deformation), hyperelastic (stores energy during deformation) and homogeneous (made of same material) myocardium. Biaxial tests have shown that passive myocardium exhibits anisotropy due to the fibre orientation and structure of myocardial tissue (Guccione <emphasis>et al.</emphasis> 1991). Here, Guccione <emphasis>et al.</emphasis> assumed transverse isotropy (behaviour is independent of all directions of loading perpendicular to one direction). For further details refer to Guccione <emphasis>et al.</emphasis> 1991.
</para>
<para>
The model was implemented in a manner that could be used for peforming finite element model simulations on the CMISS software program developed at the Bioengineering Institute, University of Auckland. Examples of how the CellML model is used can seen <ulink url="http://cmiss.bioeng.auckland.ac.nz/development/examples/5/5g/index.html">here</ulink>.
</para>
<para>
The complete original paper reference is cited below:
</para>
<para>
Passive Material Properties of Intact Ventricular Myocardium Determined From a Cylindrical Model, J. M. Guccione, A. D. McCulloch, and L. K. Waldman, 1991. <ulink url="http://scitation.aip.org/ASMEJournals/Biomechanical/?jsessionid=1586371077146625325">
<emphasis>Journal of Biomechanical Engineering</emphasis>
</ulink>, 113, 42-55. <ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=2020175&dopt=Abstract">PubMed ID: 2020175</ulink> </para>
</sect1>
</article>
</documentation>
<!-- Global units -->
<units name="strain">
<unit units="dimensionless"/>
</units>
<units name="stress">
<unit units="dimensionless"/>
</units>
<component name="interface" cmeta:id="interface">
<!-- Variables we expect to be set/controlled externally -->
<variable units="strain" private_interface="out" name="E11"/>
<variable units="strain" private_interface="out" name="E22"/>
<variable units="strain" private_interface="out" name="E33"/>
<variable units="strain" private_interface="out" name="E12"/>
<variable units="strain" private_interface="out" name="E13"/>
<variable units="strain" private_interface="out" name="E23"/>
<variable units="strain" private_interface="out" name="aff" initial_value="0"/>
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<variable units="strain" private_interface="out" name="ann" initial_value="0"/>
<variable units="strain" private_interface="out" name="ans" initial_value="0"/>
<variable units="strain" private_interface="out" name="ass" initial_value="0"/>
<variable units="strain" private_interface="out" name="bff" initial_value="0"/>
<variable units="strain" private_interface="out" name="bfn" initial_value="0"/>
<variable units="strain" private_interface="out" name="bfs" initial_value="0"/>
<variable units="strain" private_interface="out" name="bnn" initial_value="0"/>
<variable units="strain" private_interface="out" name="bns" initial_value="0"/>
<variable units="strain" private_interface="out" name="bss" initial_value="0"/>
<variable units="strain" private_interface="out" name="kff" initial_value="0"/>
<variable units="strain" private_interface="out" name="kfn" initial_value="0"/>
<variable units="strain" private_interface="out" name="kfs" initial_value="0"/>
<variable units="strain" private_interface="out" name="knn" initial_value="0"/>
<variable units="strain" private_interface="out" name="kns" initial_value="0"/>
<variable units="strain" private_interface="out" name="kss" initial_value="0"/>
<!-- Variables we want to make available externally -->
<variable units="stress" public_interface="out" private_interface="in" name="Tdev11"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev22"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev33"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev12"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev13"/>
<variable units="stress" public_interface="out" private_interface="in" name="Tdev23"/>
</component> <!--interface-->
<component name="equations" cmeta:id="equations">
<!-- Inputs -->
<variable units="strain" public_interface="in" name="E11"/>
<variable units="strain" public_interface="in" name="E22"/>
<variable units="strain" public_interface="in" name="E33"/>
<variable units="strain" public_interface="in" name="E12"/>
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<variable units="strain" public_interface="in" name="E23"/>
<variable units="strain" public_interface="in" name="aff"/>
<variable units="strain" public_interface="in" name="afn"/>
<variable units="strain" public_interface="in" name="afs"/>
<variable units="strain" public_interface="in" name="ann"/>
<variable units="strain" public_interface="in" name="ans"/>
<variable units="strain" public_interface="in" name="ass"/>
<variable units="strain" public_interface="in" name="bff"/>
<variable units="strain" public_interface="in" name="bfn"/>
<variable units="strain" public_interface="in" name="bfs"/>
<variable units="strain" public_interface="in" name="bnn"/>
<variable units="strain" public_interface="in" name="bns"/>
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<variable units="strain" public_interface="in" name="knn"/>
<variable units="strain" public_interface="in" name="kns"/>
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<!-- Outputs computed here -->
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<variable units="stress" public_interface="out" name="Tdev22"/>
<variable units="stress" public_interface="out" name="Tdev33"/>
<variable units="stress" public_interface="out" name="Tdev12"/>
<variable units="stress" public_interface="out" name="Tdev13"/>
<variable units="stress" public_interface="out" name="Tdev23"/>
<!-- Temporary variables -->
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<variable units="strain" name="elim22"/>
<variable units="strain" name="elim33"/>
<variable units="strain" name="elim12"/>
<variable units="strain" name="elim13"/>
<variable units="strain" name="elim23"/>
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<variable units="strain" name="etol22"/>
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<variable units="strain" name="eabs11"/>
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<variable units="strain" name="eabs33"/>
<variable units="strain" name="eabs12"/>
<variable units="strain" name="eabs13"/>
<variable units="strain" name="eabs23"/>
<variable units="strain" name="aMeabs11"/>
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<variable units="strain" name="aMeabs33"/>
<variable units="strain" name="aMeabs12"/>
<variable units="strain" name="aMeabs13"/>
<variable units="strain" name="aMeabs23"/>
<variable units="strain" name="mbffM1"/>
<variable units="strain" name="mbfnM1"/>
<variable units="strain" name="mbfsM1"/>
<variable units="strain" name="mbnnM1"/>
<variable units="strain" name="mbnsM1"/>
<variable units="strain" name="mbssM1"/>
<variable units="strain" name="p11"/>
<variable units="strain" name="p12"/>
<variable units="strain" name="p13"/>
<variable units="strain" name="p22"/>
<variable units="strain" name="p23"/>
<variable units="strain" name="p33"/>
<variable units="strain" name="q11"/>
<variable units="strain" name="q12"/>
<variable units="strain" name="q13"/>
<variable units="strain" name="q22"/>
<variable units="strain" name="q23"/>
<variable units="strain" name="q33"/>
<!-- Calculating strain limits for where the slope-extrapolation starts-->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="elim11_calculation_eq">
<apply id="elim11_calculation">
<eq/>
<ci>elim11</ci>
<apply>
<times/>
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<ci>aff</ci>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="elim22_calculation_eq">
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="elim33_calculation_eq">
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<apply>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="elim12_calculation_eq">
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<apply>
<times/>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="elim13_calculation_eq">
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<apply>
<times/>
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<ci>afn</ci>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="elim23_calculation_eq">
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<eq/>
<ci>elim23</ci>
<apply>
<times/>
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<ci>ans</ci>
</apply>
</apply>
</math>
<!-- Calculating strain limits for where the slope-extrapolation starts-->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="etol11_calculation_eq">
<apply id="etol11_calculation">
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="etol22_calculation_eq">
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<eq/>
<ci>etol22</ci>
<apply>
<minus/>
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<ci>elim22</ci>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="etol33_calculation_eq">
<apply id="etol33_calculation">
<eq/>
<ci>etol33</ci>
<apply>
<minus/>
<ci>ann</ci>
<ci>elim33</ci>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="etol12_calculation_eq">
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<eq/>
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<apply>
<minus/>
<ci>afs</ci>
<ci>elim12</ci>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="etol13_calculation_eq">
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<eq/>
<ci>etol13</ci>
<apply>
<minus/>
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<ci>elim13</ci>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="etol23_calculation_eq">
<apply id="etol23_calculation">
<eq/>
<ci>etol23</ci>
<apply>
<minus/>
<ci>ans</ci>
<ci>elim23</ci>
</apply>
</apply>
</math>
<!-- Calculation of Signs -->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="signE11_calculation_eq">
<apply id="signE11_calculation">
<eq/>
<ci>signE11</ci>
<piecewise>
<piece>
<cn cellml:units="dimensionless">1.0</cn>
<apply>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="signE22_calculation_eq">
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="signE33_calculation_eq">
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<otherwise>
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<!-- Calculation of altered constants -->
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<eq/>
<ci>mbffM1</ci>
<apply>
<minus/>
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<minus/>
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<apply>
<minus/>
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<apply>
<minus/>
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<ci>bnn</ci>
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<minus/>
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<ci>bns</ci>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="mbssM1_calculation_eq">
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<apply>
<minus/>
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<ci>bss</ci>
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<!-- Calculation of Absolute values -->
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<apply id="eabs11_calculation">
<eq/>
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<apply>
<times/>
<ci>signE11</ci>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="eabs22_calculation_eq">
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<apply>
<times/>
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<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="eabs33_calculation_eq">
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<apply>
<times/>
<ci>signE33</ci>
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<apply>
<times/>
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<times/>
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<ci>eabs23</ci>
<apply>
<times/>
<ci>signE23</ci>
<ci>E23</ci>
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<!-- Calculation of aMAbs values -->
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<apply>
<minus/>
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<ci>eabs12</ci>
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<!-- Calculation of p&q values values -->
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<cn cellml:units="dimensionless">-1.0</cn>
<ci>bff</ci>
</apply>
</apply>
<ci>kff</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2</cn>
<apply>
<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>bff</ci>
<ci>elim11</ci>
<ci>signE11</ci>
<apply>
<power/>
<ci>etol11</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bff</ci>
<apply>
<power/>
<ci>elim11</ci>
<cn cellml:units="dimensionless">2.0</cn>
</apply>
<ci>mbffM1</ci>
<apply>
<power/>
<ci>etol11</ci>
<cn cellml:units="dimensionless">-2.0</cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="q11_calculation_eq">
<apply id="q11_calculation">
<eq/>
<ci>q11</ci>
<apply>
<minus/>
<apply>
<times/>
<apply>
<power/>
<ci>etol11</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bff</ci>
</apply>
</apply>
<ci>kff</ci>
<ci>elim11</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>elim11</ci>
<ci>bff</ci>
<ci>signE11</ci>
<apply>
<power/>
<ci>etol11</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>p11</ci>
<ci>elim11</ci>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="p12_calculation_eq">
<apply id="p12_calculation">
<eq/>
<ci>p12</ci>
<apply>
<times/>
<apply>
<power/>
<ci>etol12</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bfs</ci>
</apply>
</apply>
<ci>kfs</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2</cn>
<apply>
<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>bfs</ci>
<ci>elim12</ci>
<ci>signE12</ci>
<apply>
<power/>
<ci>etol12</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bfs</ci>
<apply>
<power/>
<ci>elim12</ci>
<cn cellml:units="dimensionless">2.0</cn>
</apply>
<ci>mbfsM1</ci>
<apply>
<power/>
<ci>etol12</ci>
<cn cellml:units="dimensionless">-2.0</cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="q12_calculation_eq">
<apply id="q12_calculation">
<eq/>
<ci>q12</ci>
<apply>
<minus/>
<apply>
<times/>
<apply>
<power/>
<ci>etol12</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bfs</ci>
</apply>
</apply>
<ci>kfs</ci>
<ci>elim12</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>elim12</ci>
<ci>bfs</ci>
<ci>signE12</ci>
<apply>
<power/>
<ci>etol12</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>p12</ci>
<ci>elim12</ci>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="p13_calculation_eq">
<apply id="p13_calculation">
<eq/>
<ci>p13</ci>
<apply>
<times/>
<apply>
<power/>
<ci>etol13</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bfn</ci>
</apply>
</apply>
<ci>kfn</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2</cn>
<apply>
<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>bfn</ci>
<ci>elim13</ci>
<ci>signE13</ci>
<apply>
<power/>
<ci>etol13</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bfn</ci>
<apply>
<power/>
<ci>elim13</ci>
<cn cellml:units="dimensionless">2.0</cn>
</apply>
<ci>mbfnM1</ci>
<apply>
<power/>
<ci>etol13</ci>
<cn cellml:units="dimensionless">-2.0</cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="q13_calculation_eq">
<apply id="q13_calculation">
<eq/>
<ci>q13</ci>
<apply>
<minus/>
<apply>
<times/>
<apply>
<power/>
<ci>etol13</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bfn</ci>
</apply>
</apply>
<ci>kfn</ci>
<ci>elim13</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>elim13</ci>
<ci>bfn</ci>
<ci>signE13</ci>
<apply>
<power/>
<ci>etol13</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>p13</ci>
<ci>elim13</ci>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="p22_calculation_eq">
<apply id="p22_calculation">
<eq/>
<ci>p22</ci>
<apply>
<times/>
<apply>
<power/>
<ci>etol22</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bss</ci>
</apply>
</apply>
<ci>kss</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2</cn>
<apply>
<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>bss</ci>
<ci>elim22</ci>
<ci>signE22</ci>
<apply>
<power/>
<ci>etol22</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bss</ci>
<apply>
<power/>
<ci>elim22</ci>
<cn cellml:units="dimensionless">2.0</cn>
</apply>
<ci>mbssM1</ci>
<apply>
<power/>
<ci>etol22</ci>
<cn cellml:units="dimensionless">-2.0</cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="q22_calculation_eq">
<apply id="q22_calculation">
<eq/>
<ci>q22</ci>
<apply>
<minus/>
<apply>
<times/>
<apply>
<power/>
<ci>etol22</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bss</ci>
</apply>
</apply>
<ci>kss</ci>
<ci>elim22</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>elim22</ci>
<ci>bss</ci>
<ci>signE22</ci>
<apply>
<power/>
<ci>etol22</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>p22</ci>
<ci>elim22</ci>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="p23_calculation_eq">
<apply id="p23_calculation">
<eq/>
<ci>p23</ci>
<apply>
<times/>
<apply>
<power/>
<ci>etol23</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bns</ci>
</apply>
</apply>
<ci>kns</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2</cn>
<apply>
<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>bns</ci>
<ci>elim23</ci>
<ci>signE23</ci>
<apply>
<power/>
<ci>etol23</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bns</ci>
<apply>
<power/>
<ci>elim23</ci>
<cn cellml:units="dimensionless">2.0</cn>
</apply>
<ci>mbnsM1</ci>
<apply>
<power/>
<ci>etol23</ci>
<cn cellml:units="dimensionless">-2.0</cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="q23_calculation_eq">
<apply id="q23_calculation">
<eq/>
<ci>q23</ci>
<apply>
<minus/>
<apply>
<times/>
<apply>
<power/>
<ci>etol23</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bns</ci>
</apply>
</apply>
<ci>kns</ci>
<ci>elim23</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>elim23</ci>
<ci>bns</ci>
<ci>signE23</ci>
<apply>
<power/>
<ci>etol23</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>p23</ci>
<ci>elim23</ci>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="p33_calculation_eq">
<apply id="p33_calculation">
<eq/>
<ci>p33</ci>
<apply>
<times/>
<apply>
<power/>
<ci>etol33</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bnn</ci>
</apply>
</apply>
<ci>knn</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2</cn>
<apply>
<times/>
<cn cellml:units="dimensionless">4</cn>
<ci>bnn</ci>
<ci>elim33</ci>
<ci>signE33</ci>
<apply>
<power/>
<ci>etol33</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bnn</ci>
<apply>
<power/>
<ci>elim33</ci>
<cn cellml:units="dimensionless">2.0</cn>
</apply>
<ci>mbnnM1</ci>
<apply>
<power/>
<ci>etol33</ci>
<cn cellml:units="dimensionless">-2.0</cn>
</apply>
</apply>
</apply>
</apply>
</apply>
</math>
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="q33_calculation_eq">
<apply id="q33_calculation">
<eq/>
<ci>q33</ci>
<apply>
<minus/>
<apply>
<times/>
<apply>
<power/>
<ci>etol33</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bnn</ci>
</apply>
</apply>
<ci>knn</ci>
<ci>elim33</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>elim33</ci>
<ci>bnn</ci>
<ci>signE33</ci>
<apply>
<power/>
<ci>etol33</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<times/>
<ci>p33</ci>
<ci>elim33</ci>
</apply>
</apply>
</apply>
</math>
<!-- Calculation of Stress -->
<!-- 11 direction -->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev11_calculation_eq">
<apply id="Tdev11_calculation">
<eq/>
<ci>Tdev11</ci>
<piecewise>
<piece>
<apply>
<times/>
<apply>
<power/>
<ci>aMeabs11</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bff</ci>
</apply>
</apply>
<ci>kff</ci>
<ci>E11</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>E11</ci>
<ci>bff</ci>
<ci>signE11</ci>
<apply>
<power/>
<ci>aMeabs11</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<leq/>
<ci>eabs11</ci>
<ci>elim11</ci>
</apply>
</piece>
<otherwise>
<apply>
<plus/>
<apply>
<times/>
<ci>p11</ci>
<ci>E11</ci>
</apply>
<ci>q11</ci>
</apply>
</otherwise>
</piecewise>
</apply>
</math>
<!-- 22 direction -->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev22_calculation_eq">
<apply id="Tdev22_calculation">
<eq/>
<ci>Tdev22</ci>
<piecewise>
<piece>
<apply>
<times/>
<apply>
<power/>
<ci>aMeabs22</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bss</ci>
</apply>
</apply>
<ci>kss</ci>
<ci>E22</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>E22</ci>
<ci>bss</ci>
<ci>signE22</ci>
<apply>
<power/>
<ci>aMeabs22</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<leq/>
<ci>eabs22</ci>
<ci>elim22</ci>
</apply>
</piece>
<otherwise>
<apply>
<plus/>
<apply>
<times/>
<ci>p22</ci>
<ci>E22</ci>
</apply>
<ci>q22</ci>
</apply>
</otherwise>
</piecewise>
</apply>
</math>
<!-- 33 direction -->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev33_calculation_eq">
<apply id="Tdev33_calculation">
<eq/>
<ci>Tdev33</ci>
<piecewise>
<piece>
<apply>
<times/>
<apply>
<power/>
<ci>aMeabs33</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bnn</ci>
</apply>
</apply>
<ci>knn</ci>
<ci>E33</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>E33</ci>
<ci>bnn</ci>
<ci>signE33</ci>
<apply>
<power/>
<ci>aMeabs33</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<leq/>
<ci>eabs33</ci>
<ci>elim33</ci>
</apply>
</piece>
<otherwise>
<apply>
<plus/>
<apply>
<times/>
<ci>p33</ci>
<ci>E33</ci>
</apply>
<ci>q33</ci>
</apply>
</otherwise>
</piecewise>
</apply>
</math>
<!-- 12 direction -->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev12_calculation_eq">
<apply id="Tdev12_calculation">
<eq/>
<ci>Tdev12</ci>
<piecewise>
<piece>
<apply>
<times/>
<apply>
<power/>
<ci>aMeabs12</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bfs</ci>
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</apply>
<ci>kfs</ci>
<ci>E12</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>E12</ci>
<ci>bfs</ci>
<ci>signE12</ci>
<apply>
<power/>
<ci>aMeabs12</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<leq/>
<ci>eabs12</ci>
<ci>elim12</ci>
</apply>
</piece>
<otherwise>
<apply>
<plus/>
<apply>
<times/>
<ci>p12</ci>
<ci>E12</ci>
</apply>
<ci>q12</ci>
</apply>
</otherwise>
</piecewise>
</apply>
</math>
<!-- 13 direction -->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev13_calculation_eq">
<apply id="Tdev13_calculation">
<eq/>
<ci>Tdev13</ci>
<piecewise>
<piece>
<apply>
<times/>
<apply>
<power/>
<ci>aMeabs13</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bfn</ci>
</apply>
</apply>
<ci>kfn</ci>
<ci>E13</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>E13</ci>
<ci>bfn</ci>
<ci>signE13</ci>
<apply>
<power/>
<ci>aMeabs13</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<leq/>
<ci>eabs13</ci>
<ci>elim13</ci>
</apply>
</piece>
<otherwise>
<apply>
<plus/>
<apply>
<times/>
<ci>p13</ci>
<ci>E13</ci>
</apply>
<ci>q13</ci>
</apply>
</otherwise>
</piecewise>
</apply>
</math>
<!-- 23 direction -->
<math xmlns="http://www.w3.org/1998/Math/MathML" cmeta:id="Tdev23_calculation_eq">
<apply id="Tdev23_calculation">
<eq/>
<ci>Tdev23</ci>
<piecewise>
<piece>
<apply>
<times/>
<apply>
<power/>
<ci>aMeabs23</ci>
<apply>
<times/>
<cn cellml:units="dimensionless">-1.0</cn>
<ci>bns</ci>
</apply>
</apply>
<ci>kns</ci>
<ci>E23</ci>
<apply>
<plus/>
<cn cellml:units="dimensionless">2.0</cn>
<apply>
<times/>
<ci>E23</ci>
<ci>bns</ci>
<ci>signE23</ci>
<apply>
<power/>
<ci>aMeabs23</ci>
<cn cellml:units="dimensionless">-1.0</cn>
</apply>
</apply>
</apply>
</apply>
<apply>
<leq/>
<ci>eabs23</ci>
<ci>elim23</ci>
</apply>
</piece>
<otherwise>
<apply>
<plus/>
<apply>
<times/>
<ci>p23</ci>
<ci>E23</ci>
</apply>
<ci>q23</ci>
</apply>
</otherwise>
</piecewise>
</apply>
</math>
<!-- END END END END END END END END END END END END END END END -->
</component>
<!--membrane-->
<connection>
<map_components component_2="equations" component_1="interface"/>
<map_variables variable_2="E11" variable_1="E11"/>
<map_variables variable_2="E22" variable_1="E22"/>
<map_variables variable_2="E33" variable_1="E33"/>
<map_variables variable_2="E12" variable_1="E12"/>
<map_variables variable_2="E13" variable_1="E13"/>
<map_variables variable_2="E23" variable_1="E23"/>
<map_variables variable_2="Tdev11" variable_1="Tdev11"/>
<map_variables variable_2="Tdev22" variable_1="Tdev22"/>
<map_variables variable_2="Tdev33" variable_1="Tdev33"/>
<map_variables variable_2="Tdev12" variable_1="Tdev12"/>
<map_variables variable_2="Tdev13" variable_1="Tdev13"/>
<map_variables variable_2="Tdev23" variable_1="Tdev23"/>
<map_variables variable_2="aff" variable_1="aff"/>
<map_variables variable_2="afn" variable_1="afn"/>
<map_variables variable_2="afs" variable_1="afs"/>
<map_variables variable_2="ann" variable_1="ann"/>
<map_variables variable_2="ans" variable_1="ans"/>
<map_variables variable_2="ass" variable_1="ass"/>
<map_variables variable_2="bff" variable_1="bff"/>
<map_variables variable_2="bfn" variable_1="bfn"/>
<map_variables variable_2="bfs" variable_1="bfs"/>
<map_variables variable_2="bnn" variable_1="bnn"/>
<map_variables variable_2="bns" variable_1="bns"/>
<map_variables variable_2="bss" variable_1="bss"/>
<map_variables variable_2="kff" variable_1="kff"/>
<map_variables variable_2="kfn" variable_1="kfn"/>
<map_variables variable_2="kfs" variable_1="kfs"/>
<map_variables variable_2="knn" variable_1="knn"/>
<map_variables variable_2="kns" variable_1="kns"/>
<map_variables variable_2="kss" variable_1="kss"/>
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<group>
<relationship_ref relationship="encapsulation"/>
<component_ref component="interface">
<component_ref component="equations"/>
</component_ref>
</group>
<rdf:RDF>
<rdf:Seq rdf:about="rdf:#citationAuthorsSeq">
<rdf:li rdf:resource="rdf:#author1Vcard"/>
<rdf:li rdf:resource="rdf:#author2Vcard"/>
<rdf:li rdf:resource="rdf:#author3Vcard"/>
</rdf:Seq>
<rdf:Description rdf:about="#interface">
<cmeta:comment rdf:resource="rdf:#40a9732b-83b6-4d14-86a6-4cec0d5ce166"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#6dc84e7b-4c5b-4a18-ad4a-2b2ee862062a">
<vCard:ORG rdf:resource="rdf:#239f7fc9-1831-4144-aff3-86476786dd9b"/>
<vCard:EMAIL rdf:resource="rdf:#41724147-5e70-42ec-907a-14398a7f26dd"/>
<vCard:N rdf:resource="rdf:#05e625c3-c5ec-4dba-aacd-a9ee02e0a7d7"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author1Vcard">
<rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
<vCard:N rdf:resource="rdf:#author1VcardN"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#41724147-5e70-42ec-907a-14398a7f26dd">
<rdf:type rdf:resource="http://imc.org/vCard/3.0#internet"/>
<rdf:value>h.schmid@auckland.ac.nz</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="rdf:#69a8e23a-7001-4b54-a16b-66af6b879c86">
<vCard:N rdf:resource="rdf:#a664f94c-102d-466f-8c9b-e7265f42ba18"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author3VcardN">
<vCard:Given>L</vCard:Given>
<vCard:Family>Waldman</vCard:Family>
<vCard:Other>K</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#239f7fc9-1831-4144-aff3-86476786dd9b">
<vCard:Orgname>The University of Auckland</vCard:Orgname>
<vCard:Orgunit>The Bioengineering Institute</vCard:Orgunit>
</rdf:Description>
<rdf:Description rdf:about="rdf:#7bd5d993-2f89-4d5e-bd70-c0dd3b4f7cb3">
<dc:creator rdf:resource="rdf:#citationAuthorsSeq"/>
<dc:title>Passive Material Properties of Intact Ventricular Myocardium Determined From a Cylindrical Model</dc:title>
<bqs:volume>113</bqs:volume>
<bqs:first_page>42</bqs:first_page>
<bqs:Journal rdf:resource="rdf:#c8dd7446-c9f2-4bb9-8a13-73705612e558"/>
<dcterms:issued rdf:resource="rdf:#535ee7b8-fa35-42b2-9e03-07f96d666f98"/>
<bqs:last_page>55</bqs:last_page>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author2VcardN">
<vCard:Given>A</vCard:Given>
<vCard:Family>McCulloch</vCard:Family>
<vCard:Other>D</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#d955f844-4d65-4cef-9e67-62c1eed24a0b">
<rdf:value>
In this simple model we only have one component, which holds the
six equations.
</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="">
<dcterms:created rdf:resource="rdf:#1c2d92e0-c4a1-47c5-99e3-87cbfae7f410"/>
<dc:creator rdf:resource="rdf:#6dc84e7b-4c5b-4a18-ad4a-2b2ee862062a"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author2Vcard">
<rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
<vCard:N rdf:resource="rdf:#author2VcardN"/>
</rdf:Description>
<rdf:Description rdf:about="#equations">
<cmeta:comment rdf:resource="rdf:#d955f844-4d65-4cef-9e67-62c1eed24a0b"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#24eb56e0-881a-4cae-a286-f2773b808d28">
<dcterms:modified rdf:resource="rdf:#9c271aab-0d15-4216-9965-f45d86406e84"/>
<rdf:value>
Added metadata to the model.
</rdf:value>
<rdf:type rdf:resource="http://www.cellml.org/metadata/1.0#modification"/>
<cmeta:modifier rdf:resource="rdf:#69a8e23a-7001-4b54-a16b-66af6b879c86"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#9c271aab-0d15-4216-9965-f45d86406e84">
<dcterms:W3CDTF>2004-02-18</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#3711ae44-104e-43d7-9d66-78fa995ae0bd">
<bqs:subject_type>keyword</bqs:subject_type>
<rdf:value>
<rdf:Bag>
<rdf:li>constitutive material law</rdf:li>
<rdf:li>mechanical constitutive laws</rdf:li>
</rdf:Bag>
</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="rdf:#1c2d92e0-c4a1-47c5-99e3-87cbfae7f410">
<dcterms:W3CDTF>2003-12-28</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="rdf:#88cc0a0a-47ca-4676-ada7-84a18dd11a1d">
<bqs:Pubmed_id>2020175</bqs:Pubmed_id>
<bqs:JournalArticle rdf:resource="rdf:#7bd5d993-2f89-4d5e-bd70-c0dd3b4f7cb3"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#100ee02c-63ff-48e9-9b0b-e8331f701369">
<rdf:value>
This is a CellML version of the Separate Fung constitutive material law,
defining the relation between the six independent strain components
and the stress components. It is assumed that the strain components
will be controlled externally by the application using this CellML
model.
</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="rdf:#40a9732b-83b6-4d14-86a6-4cec0d5ce166">
<rdf:value>
We'll use this component as the "interface" to the model, all
other components are hidden via encapsulation in this component.
</rdf:value>
</rdf:Description>
<rdf:Description rdf:about="rdf:#c8dd7446-c9f2-4bb9-8a13-73705612e558">
<dc:title>Journal of Biomechanical Engineering</dc:title>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author3Vcard">
<rdf:type rdf:resource="http://www.cellml.org/bqs/1.0#Person"/>
<vCard:N rdf:resource="rdf:#author3VcardN"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#author1VcardN">
<vCard:Given>J</vCard:Given>
<vCard:Family>Guccione</vCard:Family>
<vCard:Other>M</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#a664f94c-102d-466f-8c9b-e7265f42ba18">
<vCard:Given>Catherine</vCard:Given>
<vCard:Family>Lloyd</vCard:Family>
<vCard:Other>M</vCard:Other>
</rdf:Description>
<rdf:Description rdf:about="rdf:#a659be0e-5325-462f-88ad-b01b796f3f4f">
<dc:subject rdf:resource="rdf:#3711ae44-104e-43d7-9d66-78fa995ae0bd"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#535ee7b8-fa35-42b2-9e03-07f96d666f98">
<dcterms:W3CDTF>1991-01-01</dcterms:W3CDTF>
</rdf:Description>
<rdf:Description rdf:about="#PoleZero">
<bqs:reference rdf:resource="rdf:#88cc0a0a-47ca-4676-ada7-84a18dd11a1d"/>
<bqs:reference rdf:resource="rdf:#a659be0e-5325-462f-88ad-b01b796f3f4f"/>
<cmeta:comment rdf:resource="rdf:#100ee02c-63ff-48e9-9b0b-e8331f701369"/>
</rdf:Description>
<rdf:Description rdf:about="rdf:#05e625c3-c5ec-4dba-aacd-a9ee02e0a7d7">
<vCard:Given>Holger</vCard:Given>
<vCard:Family>Schmid</vCard:Family>
</rdf:Description>
</rdf:RDF>
</model>
