<documentation xmlns="http://cellml.org/tmp-documentation"> <articleinfo> Minimal haemodynamic system model including ventricular interaction and valve dynamics. <author> <firstname>Geoffrey</firstname> <surname>Nunns</surname> <affiliation> <shortaffil>Bioengineering Institute, The University of Auckland</shortaffil> </affiliation> </author> </articleinfo> Model Status This CellML model will run in PCEnv and OpenCell to recreate the published results. Note that the model will not run in COR because it contains differential algebraic equations (DAEs). The CellML transaltion of the model is based on the original Matlab code obtained from the model authors. However, the CellML model will only simulate 14 heart beats, because if statements (piecewise equations) were used instead of a remainder operator to cycle the heart beat, but this number of beats can be increased if desired. <sect1 id="sec_structure"> Model Structure Abstract: Characterising circulatory dysfunction and choosing a suitable treatment is often difficult and time consuming, and can result in a deterioration in patient condition, or unsuitable therapy choices. A stable minimal model of the human cardiovascular system (CVS) is developed with the ultimate specific aim of assisting medical staff for rapid, on site modelling to assist in diagnosis and treatment. Models found in the literature simulate specific areas of the CVS with limited direct usefulness to medical staff. Others model the full CVS as a closed loop system, but they were found to be very complex, difficult to solve, or unstable. This paper develops a model that uses a minimal number of governing equations with the primary goal of accurately capturing trends in the CVS dynamics in a simple, easily solved, robust model. The model is shown to have long term stability and consistency with non-specific initial conditions as a result. An "open on pressure close on flow" valve law is created to capture the effects of inertia and the resulting dynamics of blood flow through the cardiac valves. An accurate, stable solution is performed using a method that varies the number of states in the model depending on the specific phase of the cardiac cycle, better matching the real physiological conditions. Examples of results include a 9% drop in cardiac output when increasing the thoracic pressure from -4 to 0 mmHg, and an increase in blood pressure from 120/80 to 165/130 mmHg when the systemic resistance is doubled. These results show that the model adequately provides appropriate magnitudes and trends that are in agreement with existing data for a variety of physiologically verified test cases simulating human CVS function. The complete original paper reference is cited below: Minimal haemodynamic system model including ventricular interaction and valve dynamics, Bram W Smith, J Geoffrey Chase, Roger I Nokes, Geoffrey M Shaw, Graeme Wake, 2004, Medical Engineering and Physics , 26, 131-139. PubMed ID: 15036180 <informalfigure float="0" id="fig_reaction_diagram"> <mediaobject> <imageobject> <objectinfo> model diagram </objectinfo> <imagedata fileref="shaw_2007.png" /> </imageobject> </mediaobject> Schematic diagram of the minimal closed loop model of the cardiovascular system. </informalfigure> </sect1> </documentation>
<rdf:RDF xmlns:cmeta="http://www.cellml.org/metadata/1.0#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:j.0="http://www.example.org/foo/bar/" xmlns="http://www.w3.org/2001/vcard-rdf/3.0#" xmlns:j.1="http://www.cellml.org/bqs/1.0#" xmlns:j.2="http://imc.org/vCard/3.0#" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:j.3="http://purl.org/dc/terms/" xmlns:j.4="http://www.cellml.org/metadata/simulation/1.0#"> <rdf:Description rdf:about="http://www.cellml.org/cellml/1.0"> <j.3:created rdf:parseType="Resource"> <j.3:W3CDTF xml:lang="en">2009-12-01</j.3:W3CDTF> </j.3:created> <dc:creator rdf:parseType="Resource"> <N rdf:parseType="Resource"> <Other xml:lang="en">Rogan</Other> <Given xml:lang="en">Geoffrey</Given> <Family xml:lang="en">Nunns</Family> </N> <ORG rdf:parseType="Resource"> <Orgunit xml:lang="en">Auckland Bioengineering Institute</Orgunit> <Orgname xml:lang="en">The University of Auckland</Orgname> </ORG> <EMAIL> <j.2:internet> <rdf:value xml:lang="en">gnunns1@jhu.edu</rdf:value> </j.2:internet> </EMAIL> </dc:creator> </rdf:Description> <j.0:bazfoo/> <rdf:Description rdf:about="http://www.cellml.org/cellml/1.0#smith_chase_nokes_shaw_wake_2004"> <j.4:simulation rdf:parseType="Resource"> <j.4:boundIntervals rdf:parseType="Collection"> <rdf:Description> <j.4:maximumStepSize xml:lang="en">0.1</j.4:maximumStepSize> <j.4:endingValue xml:lang="en">2</j.4:endingValue> <j.4:nonstandard-pointDensity xml:lang="en">10000</j.4:nonstandard-pointDensity> </rdf:Description> </j.4:boundIntervals> </j.4:simulation> <cmeta:bio_entity xml:lang="en">heart</cmeta:bio_entity> <cmeta:species xml:lang="en">Human</cmeta:species> <cmeta:comment rdf:parseType="Resource"> <rdf:value xml:lang="en">This is the CellML description of Smith et al.'s 2004 minimal haemodynamic system model including valve mechanics and ventricular interactions</rdf:value> <dc:creator rdf:parseType="Resource"> Geoffrey Nunns </dc:creator> </cmeta:comment> <dc:title xml:lang="en">The Smith et al. 2004 minimal haemodynamic system model including valve mechanics and ventricular interactions</dc:title> <j.1:reference rdf:parseType="Resource"> <dc:subject rdf:parseType="Resource"> <rdf:value> <rdf:Bag> <rdf:li xml:lang="en">cardiovascular circulation</rdf:li> </rdf:Bag> </rdf:value> <j.1:subject_type xml:lang="en">keyword</j.1:subject_type> </dc:subject> </j.1:reference> <j.1:reference rdf:parseType="Resource"> <j.1:Pubmed_id xml:lang="en">15036180</j.1:Pubmed_id> <j.1:JournalArticle rdf:parseType="Resource"> <j.1:first_page xml:lang="en">131</j.1:first_page> <j.3:issued rdf:parseType="Resource"> <j.3:W3CDTF xml:lang="en">2004-03-26</j.3:W3CDTF> </j.3:issued> <dc:creator> <rdf:Seq> <rdf:li rdf:parseType="Resource"> <j.1:Person rdf:parseType="Resource"> <N rdf:parseType="Resource"> <Other xml:lang="en">W</Other> <Given xml:lang="en">Bram</Given> <Family xml:lang="en">Smith</Family> </N> </j.1:Person> </rdf:li> <rdf:li rdf:parseType="Resource"> <j.1:Person rdf:parseType="Resource"> <N rdf:parseType="Resource"> <Other xml:lang="en">Geoffrey</Other> <Given xml:lang="en">J</Given> <Family xml:lang="en">Chase</Family> </N> </j.1:Person> </rdf:li> <rdf:li rdf:parseType="Resource"> <j.1:Person rdf:parseType="Resource"> <N rdf:parseType="Resource"> <Other xml:lang="en">I</Other> <Given xml:lang="en">Roger</Given> <Family xml:lang="en">Nokes</Family> </N> </j.1:Person> </rdf:li> <rdf:li rdf:parseType="Resource"> <j.1:Person rdf:parseType="Resource"> <N rdf:parseType="Resource"> <Other xml:lang="en">M</Other> <Given xml:lang="en">Geoffrey</Given> <Family xml:lang="en">Shaw</Family> </N> </j.1:Person> </rdf:li> <rdf:li rdf:parseType="Resource"> <j.1:Person rdf:parseType="Resource"> <N rdf:parseType="Resource"> <Given xml:lang="en">Graeme</Given> <Family xml:lang="en">Wake</Family> </N> </j.1:Person> </rdf:li> </rdf:Seq> </dc:creator> <j.1:Journal rdf:parseType="Resource"> <dc:title xml:lang="en">Medical Engineering and Physics</dc:title> </j.1:Journal> <dc:title xml:lang="en">Minimal haemodynamic system model including ventricular interaction and valve dynamics.</dc:title> <j.1:last_page xml:lang="en">139</j.1:last_page> <j.1:volume xml:lang="en">26</j.1:volume> </j.1:JournalArticle> </j.1:reference> </rdf:Description> </rdf:RDF>
A minimal model of the cardivascular system
Chris Thompson
Simple Cardiovascular Model
smith_chase_nokes_shaw_wake_2004
Fluid volume
Fluid Flow Rate
Fluid Pressure
Volume of spatial region
Lumen of aorta
Lumen of pulmonary trunk
Set of lumens of vena cava trunks
Lumen of pulmonary arterial tree
Lumen of pulmonary venous tree organ
Cavity of right ventricle
Cavity of left ventricle
Portion of pericardial fluid
Cardiac septum
Pericardium
Portion of blood
Wall of left ventricle
Wall of right ventricle
A physical entity that is defined by linking multiple physical entity terms with structural relations.
A mathematical and/or logical expression that determines the value of a data structure.
An element of a SemSim model that relates to the mathematical or logical representation of physical quantities or qualities.
A physical dependency defined in a specific SemSim model.
A physical entity defined in a specific SemSim model.
A physical process defined in a specific SemSim model.
A physical property defined in a specific SemSim model.
A computational element that stores numerical and/or logical information related to the model.
A data structure whose value is limited to decimals.
A data structure whose value is limited to the whole numbers.
A data structure unique to JSim's Mathematical Modeling Language (MML) used to set one or more model input values at runtime.
A quantitative relation between values of physical properties that occurs over a span of time and that determines the time-course of a physical process.
A physics analytical entity that represents the material, energy, and/or information content of material or immaterial, space-occupying continuants.
A SemSim model element used to represent, in whole or in part, the physical meaning of a computational element.
A physical processural entity that is a flow, or the control of a flow, of energy and/or information amongst physical continuants that are participants in the physical process.
A physical attribute of an instance of a physical entity, an occurrence of a physical process, or a physical dependency that has a quantitative value that is measurable by a physical device or method, or is computable from such values.
A data structure whose value is limited to the real numbers.
A physical dependency that is defined using a term in an external knowledge resource.
A physical entity that is defined using a term from an external knowledge resource.
A physical process that is defined using a term in an external knowledge resource.
A physical property that is defined using a term in an external knowledge resource.
A logical expression that constrains, but does not define, the value of a data structure. Example: x > 0.
A computational or physical element of a SemSim model.
A grouping of one or more data structures that comprise a computationally or physically distinct part of a model or submodel.
A physical quantity chosen as a standard in terms of which other physical quantities may be expressed.
Blood flow through aortic valve
Blood flow from the left ventricle through the atrial valve to the aorta.
Blood flow through mitral valve
Blood flow from the pulmonary vein through the mitral valve to the left ventricle.
Blood flow through pulmonary valve
Blood flow from the right ventricle through the pulmonary valve to the pulmonary artery.
Blood flow through tricuspid valve
Blood flow from the vena cava through the tricuspid valve to the right ventricle.
Pulmonary circulatory blood flow
Blood flow from the pulmonary artery through the pulmonary circulatory system to the pulmonary vein.
Systemic circulatory blood flow
Blood flow from the aorta through systemic system to the vena cava.
Aortic end systolic elastance
Pressure in systemic arteries
Left ventricular blood pressure
Blood Pressure in the Vena Cava
Blood flow through atrial valve
Atrial valve fluid resistance
Systemic circulatory fluid resistance
Volume of blood contained in aorta
Aortic blood volume at zero pressure
Time
<apply> <eq /> <ci>P_ao</ci> <apply> <times /> <ci>E_es_ao</ci> <apply> <minus /> <ci>V_ao</ci> <ci>V_d_ao</ci> </apply> </apply> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>V_ao</ci> </apply> <piecewise> <piece> <apply> <minus /> <ci>Q_sys</ci> </apply> <apply> <lt /> <ci>Q_av</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </piece> <otherwise> <apply> <minus /> <ci>Q_av</ci> <ci>Q_sys</ci> </apply> </otherwise> </piecewise> </apply>
Gaussian Curve Magnitude
Gaussian Curve Width
Gaussian Curve Delay
Heart beat driver function
Length of Heart Contraction
Contraction Time Elapsed
<apply> <eq /> <ci>tau</ci> <piecewise> <piece> <ci>time</ci> <apply> <leq /> <ci>time</ci> <ci>period</ci> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <ci>period</ci> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">3</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">3</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">4</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">4</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">5</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">5</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">6</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">6</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">7</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">7</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">8</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">8</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">9</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">9</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">10</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">10</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">11</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">11</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">12</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">12</cn> </apply> </apply> <apply> <leq /> <ci>time</ci> <apply> <times /> <ci>period</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">13</cn> </apply> </apply> </piece> </piecewise> </apply> <apply> <eq /> <ci>e_t</ci> <apply> <times /> <ci>A</ci> <apply> <exp /> <apply> <times /> <apply> <minus /> <ci>B</ci> </apply> <apply> <power /> <apply> <minus /> <ci>tau</ci> <ci>C</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">2</cn> </apply> </apply> </apply> </apply> </apply>
Atrial valve fluid inertance
Mitrial valve fluid inertance
Pulmonary valve fluid inertance
Tricuspid valve fluid inertance
Blood pressure in the pulmonary artery
Pulmonary venous blood pressure
Right ventricular blood pressure
Blood flow through mitrial valve
Pulmonary blood flow
Blood flow across tricuspid valve
Mitrial valve fluid resistance
Pulmonary circulatory fluid resistance
Pulmonary valve fluid resistance
Tricuspid valve fluid resistance
<apply> <eq /> <ci>Q_sys</ci> <apply> <divide /> <apply> <minus /> <ci>P_ao</ci> <ci>P_vc</ci> </apply> <ci>R_sys</ci> </apply> </apply> <apply> <eq /> <ci>Q_pul</ci> <apply> <divide /> <apply> <minus /> <ci>P_pa</ci> <ci>P_pu</ci> </apply> <ci>R_pul</ci> </apply> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>Q_mt</ci> </apply> <piecewise> <piece> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second2">0</cn> <apply> <and /> <apply> <lt /> <apply> <minus /> <ci>P_pu</ci> <ci>P_lv</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="kPa">0</cn> </apply> <apply> <lt /> <ci>Q_mt</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </apply> </piece> <otherwise> <apply> <divide /> <apply> <minus /> <apply> <minus /> <ci>P_pu</ci> <ci>P_lv</ci> </apply> <apply> <times /> <ci>Q_mt</ci> <ci>R_mt</ci> </apply> </apply> <ci>L_mt</ci> </apply> </otherwise> </piecewise> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>Q_av</ci> </apply> <piecewise> <piece> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second2">0</cn> <apply> <and /> <apply> <lt /> <apply> <minus /> <ci>P_lv</ci> <ci>P_ao</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="kPa">0</cn> </apply> <apply> <lt /> <ci>Q_av</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </apply> </piece> <otherwise> <apply> <divide /> <apply> <minus /> <apply> <minus /> <ci>P_lv</ci> <ci>P_ao</ci> </apply> <apply> <times /> <ci>Q_av</ci> <ci>R_av</ci> </apply> </apply> <ci>L_av</ci> </apply> </otherwise> </piecewise> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>Q_tc</ci> </apply> <piecewise> <piece> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second2">0</cn> <apply> <and /> <apply> <lt /> <apply> <minus /> <ci>P_vc</ci> <ci>P_rv</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="kPa">0</cn> </apply> <apply> <lt /> <ci>Q_tc</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </apply> </piece> <otherwise> <apply> <divide /> <apply> <minus /> <apply> <minus /> <ci>P_vc</ci> <ci>P_rv</ci> </apply> <apply> <times /> <ci>Q_tc</ci> <ci>R_tc</ci> </apply> </apply> <ci>L_tc</ci> </apply> </otherwise> </piecewise> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>Q_pv</ci> </apply> <piecewise> <piece> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second2">0</cn> <apply> <and /> <apply> <lt /> <apply> <minus /> <ci>P_rv</ci> <ci>P_pa</ci> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="kPa">0</cn> </apply> <apply> <lt /> <ci>Q_pv</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </apply> </piece> <otherwise> <apply> <divide /> <apply> <minus /> <apply> <minus /> <ci>P_rv</ci> <ci>P_pa</ci> </apply> <apply> <times /> <ci>Q_pv</ci> <ci>R_pv</ci> </apply> </apply> <ci>L_pv</ci> </apply> </otherwise> </piecewise> </apply>
Thoracic cavity pressure
Total blood volume in the body
Left ventricular free wall end systolic pressure
End diastolic pressure across the left ventricular wall
End diastolic pressure across the left ventricular free wall
Left ventricular freewall end systolic blood pressure
Pressure across the left ventricular freewall
Pericardial blood volume
Volume of blood contained in left ventricle
Volume of the free wall of the left ventricle
Septal wall volume
Left Ventricular Freewall Diastolic Exponential Parameter
<apply> <eq /> <ci>V_lvf</ci> <apply> <minus /> <ci>V_lv</ci> <ci>V_spt</ci> </apply> </apply> <apply> <eq /> <ci>P_lvf</ci> <apply> <plus /> <apply> <times /> <ci>e_t</ci> <ci>P_es_lvf</ci> </apply> <apply> <times /> <apply> <minus /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <ci>e_t</ci> </apply> <ci>P_ed_lvf</ci> </apply> </apply> </apply> <apply> <eq /> <ci>P_lv</ci> <apply> <plus /> <ci>P_lvf</ci> <ci>P_peri</ci> </apply> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>V_lv</ci> </apply> <piecewise> <piece> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> <apply> <and /> <apply> <lt /> <ci>Q_mt</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> <apply> <lt /> <ci>Q_av</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>Q_av</ci> </apply> <apply> <lt /> <ci>Q_mt</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </piece> <piece> <ci>Q_mt</ci> <apply> <lt /> <ci>Q_av</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </piece> <otherwise> <apply> <minus /> <ci>Q_mt</ci> <ci>Q_av</ci> </apply> </otherwise> </piecewise> </apply>
Left ventricular free wall volume at end diastolic pressure
Left ventricular free wall volume at end zero pressure
<apply> <eq /> <ci>P_es_lvf</ci> <apply> <times /> <ci>E_es_lvf</ci> <apply> <minus /> <ci>V_lvf</ci> <ci>V_d_lvf</ci> </apply> </apply> </apply> <apply> <eq /> <ci>P_ed_lvf</ci> <apply> <times /> <ci>P_0_lvf</ci> <apply> <minus /> <apply> <exp /> <apply> <times /> <ci>lambda_lvf</ci> <apply> <minus /> <ci>V_lvf</ci> <ci>V_0_lvf</ci> </apply> </apply> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> </apply>
End diastolic pressure across the pericardium
Pressure across the pericardial wall
Pericardial volume at end diastolic pressure
Volume of the pericardium
Right ventricle blood volume
Pericardial Diastolic Exponential Parameter
<apply> <eq /> <ci>V_pcd</ci> <apply> <plus /> <ci>V_lv</ci> <ci>V_rv</ci> </apply> </apply> <apply> <eq /> <ci>P_pcd</ci> <apply> <times /> <ci>P_0_pcd</ci> <apply> <minus /> <apply> <exp /> <apply> <times /> <ci>lambda_pcd</ci> <apply> <minus /> <ci>V_pcd</ci> <ci>V_0_pcd</ci> </apply> </apply> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> </apply> <apply> <eq /> <ci>P_peri</ci> <apply> <plus /> <ci>P_pcd</ci> <ci>P_th</ci> </apply> </apply>
Pulmonary arterial end systolic elastance
Pulmonary arterial blood volume at zero pressure
Pulmonary arterial blood volume
<apply> <eq /> <ci>P_pa</ci> <apply> <plus /> <apply> <times /> <ci>E_es_pa</ci> <apply> <minus /> <ci>V_pa</ci> <ci>V_d_pa</ci> </apply> </apply> <ci>P_th</ci> </apply> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>V_pa</ci> </apply> <piecewise> <piece> <apply> <minus /> <ci>Q_pul</ci> </apply> <apply> <lt /> <ci>Q_pv</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </piece> <otherwise> <apply> <minus /> <ci>Q_pv</ci> <ci>Q_pul</ci> </apply> </otherwise> </piecewise> </apply>
End systolic pulmonary vein elastance
Pulmonary venous blood volume
<apply> <eq /> <ci>P_pu</ci> <apply> <plus /> <apply> <times /> <ci>E_es_pu</ci> <apply> <minus /> <ci>V_pu</ci> <ci>V_d_pu</ci> </apply> </apply> <ci>P_th</ci> </apply> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>V_pu</ci> </apply> <piecewise> <piece> <ci>Q_pul</ci> <apply> <lt /> <ci>Q_mt</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </piece> <otherwise> <apply> <minus /> <ci>Q_pul</ci> <ci>Q_mt</ci> </apply> </otherwise> </piecewise> </apply>
Right ventricular free wall end systolic elastance
Right ventricular free wall end diastolic pressure
Right ventricular free wall end systolic pressure
Pressure across the right ventricular free wall
Right ventricular free wall volume
Right Ventricular Freewall Diastolic Exponential Parameter
<apply> <eq /> <ci>V_rvf</ci> <apply> <plus /> <ci>V_rv</ci> <ci>V_spt</ci> </apply> </apply> <apply> <eq /> <ci>P_rvf</ci> <apply> <plus /> <apply> <times /> <ci>e_t</ci> <ci>P_es_rvf</ci> </apply> <apply> <times /> <apply> <minus /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <ci>e_t</ci> </apply> <ci>P_ed_rvf</ci> </apply> </apply> </apply> <apply> <eq /> <ci>P_rv</ci> <apply> <plus /> <ci>P_rvf</ci> <ci>P_peri</ci> </apply> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>V_rv</ci> </apply> <piecewise> <piece> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> <apply> <and /> <apply> <lt /> <ci>Q_tc</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> <apply> <lt /> <ci>Q_pv</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </apply> </piece> <piece> <apply> <minus /> <ci>Q_pv</ci> </apply> <apply> <lt /> <ci>Q_tc</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </piece> <piece> <ci>Q_tc</ci> <apply> <lt /> <ci>Q_pv</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </piece> <otherwise> <apply> <minus /> <ci>Q_tc</ci> <ci>Q_pv</ci> </apply> </otherwise> </piecewise> </apply>
Right ventricular free wall volume at end diastolic pressure
Right ventricular free wall volume at zero pressure
<apply> <eq /> <ci>P_es_rvf</ci> <apply> <times /> <ci>E_es_rvf</ci> <apply> <minus /> <ci>V_rvf</ci> <ci>V_d_rvf</ci> </apply> </apply> </apply> <apply> <eq /> <ci>P_ed_rvf</ci> <apply> <times /> <ci>P_0_rvf</ci> <apply> <minus /> <apply> <exp /> <apply> <times /> <ci>lambda_rvf</ci> <apply> <minus /> <ci>V_rvf</ci> <ci>V_0_rvf</ci> </apply> </apply> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> </apply>
Left ventricular free wall end systolic elastance
Septal end systolic elastance
Septal wall end diastolic pressure
Septal wall volume at end diastolic pressure
Septal wall volume at zero pressure
Septal Wall Diastolic Exponential Parameter
<apply> <eq /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="kPa">0</cn> <apply> <plus /> <apply> <minus /> <apply> <minus /> <apply> <plus /> <apply> <times /> <ci>e_t</ci> <ci>E_es_spt</ci> <apply> <minus /> <ci>V_spt</ci> <ci>V_d_spt</ci> </apply> </apply> <apply> <times /> <apply> <minus /> <ci>one</ci> <ci>e_t</ci> </apply> <ci>P_0_spt</ci> <apply> <minus /> <apply> <exp /> <apply> <times /> <ci>lambda_spt</ci> <apply> <minus /> <ci>V_spt</ci> <ci>V_0_spt</ci> </apply> </apply> </apply> <ci>one</ci> </apply> </apply> </apply> <apply> <times /> <ci>e_t</ci> <ci>E_es_lvf</ci> <apply> <minus /> <ci>V_lv</ci> <ci>V_spt</ci> </apply> </apply> </apply> <apply> <times /> <apply> <minus /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <ci>e_t</ci> </apply> <ci>P_0_lvf</ci> <apply> <minus /> <apply> <exp /> <apply> <times /> <ci>lambda_lvf</ci> <apply> <minus /> <ci>V_lv</ci> <ci>V_spt</ci> </apply> </apply> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> </apply> <apply> <times /> <ci>e_t</ci> <ci>E_es_rvf</ci> <apply> <plus /> <ci>V_rv</ci> <ci>V_spt</ci> </apply> </apply> <apply> <times /> <apply> <minus /> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> <ci>e_t</ci> </apply> <ci>P_0_rvf</ci> <apply> <minus /> <apply> <exp /> <apply> <times /> <ci>lambda_rvf</ci> <apply> <plus /> <ci>V_rv</ci> <ci>V_spt</ci> </apply> </apply> </apply> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="dimensionless">1</cn> </apply> </apply> </apply> </apply>
Vena cava end systolic elastance
Vena cava volume at zero pressure
Vena cava blood volume
<apply> <eq /> <ci>P_vc</ci> <apply> <times /> <ci>E_es_vc</ci> <apply> <minus /> <ci>V_vc</ci> <ci>V_d_vc</ci> </apply> </apply> </apply> <apply> <eq /> <apply> <diff /> <bvar> <ci>time</ci> </bvar> <ci>V_vc</ci> </apply> <piecewise> <piece> <ci>Q_sys</ci> <apply> <lt /> <ci>Q_tc</ci> <cn xmlns:cellml="http://www.cellml.org/cellml/1.0#" cellml:units="mL_per_second">0</cn> </apply> </piece> <otherwise> <apply> <minus /> <ci>Q_sys</ci> <ci>Q_tc</ci> </apply> </otherwise> </piecewise> </apply>