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<title>Minimal haemodynamic system model including ventricular interaction and valve dynamics.</title>
<author>
<firstname>Geoffrey</firstname>
<surname>Nunns</surname>
<affiliation>
<shortaffil>Bioengineering Institute, The University of Auckland</shortaffil>
</affiliation>
</author>
</articleinfo>
<section id="sec_status">
<title>Model Status</title>
<para>This 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.</para>
</section>
<sect1 id="sec_structure">
<title>Model Structure</title>
<para>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.</para>
<para>The complete original paper reference is cited below:</para>
<para>
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,
<emphasis>Medical Engineering and Physics</emphasis>
, 26, 131-139.
<ulink url="http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=15036180&dopt=Abstract">PubMed ID: 15036180</ulink>
</para>
<informalfigure float="0" id="fig_reaction_diagram">
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<caption>Schematic diagram of the minimal closed loop model of the cardiovascular system.</caption>
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A minimal model of the cardivascular system
Chris Thompson
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A physical property defined in a specific SemSim model.
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A grouping of one or more data structures that comprise a computationally or physically distinct part of a model or submodel.
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Blood flow through mitrial valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
out
Pulmonary blood flow
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
out
Blood flow through pulmonary valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
out
Systemic circulatory blood flow
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
out
Blood flow across tricuspid valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Atrial valve fluid resistance
false
in
Mitrial valve fluid resistance
false
in
Pulmonary circulatory fluid resistance
false
in
Pulmonary valve fluid resistance
false
in
Systemic circulatory fluid resistance
false
in
Tricuspid valve fluid resistance
Time
false
http://bhi.washington.edu/OPB#OPB01023
in
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
Lumen of aorta
http://purl.org/sig/ont/fma/fma12239
Lumen of pulmonary trunk
http://purl.org/sig/ont/fma/fma14167
Set of lumens of vena cava trunks
http://purl.org/sig/ont/fma/fma280059
Lumen of pulmonary arterial tree
http://purl.org/sig/ont/fma/fma52449
Lumen of pulmonary venous tree organ
http://purl.org/sig/ont/fma/fma52455
Cavity of right ventricle
http://purl.org/sig/ont/fma/fma9291
Cavity of left ventricle
http://purl.org/sig/ont/fma/fma9466
Portion of pericardial fluid
http://purl.org/sig/ont/fma/fma9887
heart_parameters
false
out
Atrial valve fluid inertance
false
out
Mitrial valve fluid inertance
false
out
Pulmonary valve fluid inertance
false
out
Tricuspid valve fluid inertance
false
out
Thoracic cavity pressure
false
heart_parameters_R_av
out
Atrial valve fluid resistance
false
out
Mitrial valve fluid resistance
false
heart_parameters_R_pul
out
Pulmonary circulatory fluid resistance
false
heart_parameters_R_pv
out
Pulmonary valve fluid resistance
false
heart_parameters_R_sys
out
Systemic circulatory fluid resistance
false
heart_parameters_R_tc
out
Tricuspid valve fluid resistance
false
out
Total blood volume in the body
left_ventricle
encapsulation
false
in
out
Left ventricular free wall end systolic pressure
false
in
out
End diastolic pressure across the left ventricular wall
false
in
Pressure in systemic arteries
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
End diastolic pressure across the left ventricular free wall
false
in
Left ventricular freewall end systolic blood pressure
false
left_ventricle_P_lv
out
Left ventricular blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
out
Pressure across the left ventricular freewall
false
in
Pericardial blood volume
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Pulmonary venous blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Blood flow through atrial valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Blood flow through mitrial valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Atrial valve fluid resistance
false
in
Mitrial valve fluid resistance
false
left_ventricle_V_lv
out
Volume of blood contained in left ventricle
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
out
out
Volume of the free wall of the left ventricle
Volume of spatial region
http://bhi.washington.edu/OPB#OPB_01061
false
in
Septal wall volume
Volume of spatial region
http://bhi.washington.edu/OPB#OPB_01061
false
in
Heart beat driver function
false
in
out
Left Ventricular Freewall Diastolic Exponential Parameter
Time
false
http://bhi.washington.edu/OPB#OPB01023
in
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
lvf_calculator
false
out
Left ventricular free wall end systolic pressure
false
out
End diastolic pressure across the left ventricular wall
false
out
End diastolic pressure across the left ventricular free wall
false
out
Left ventricular freewall end systolic blood pressure
false
Left ventricular free wall volume at end diastolic pressure
false
Left ventricular free wall volume at end zero pressure
false
in
Volume of the free wall of the left ventricle
Volume of spatial region
http://bhi.washington.edu/OPB#OPB_01061
false
out
Left Ventricular Freewall Diastolic Exponential Parameter
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
pericardium
false
End diastolic pressure across the pericardium
false
out
Pressure across the pericardial wall
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
out
Pericardial blood volume
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Thoracic cavity pressure
false
Pericardial volume at end diastolic pressure
false
in
Volume of blood contained in left ventricle
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
out
Volume of the pericardium
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
in
Right ventricle blood volume
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
Pericardial Diastolic Exponential Parameter
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
pulmonary_artery
false
Pulmonary arterial end systolic elastance
false
out
Blood pressure in the pulmonary artery
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Pulmonary venous blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Right ventricular blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Thoracic cavity pressure
false
in
Pulmonary blood flow
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Blood flow through pulmonary valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Pulmonary circulatory fluid resistance
false
in
Pulmonary valve fluid resistance
false
Pulmonary arterial blood volume at zero pressure
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
pulmonary_artery_V_pa
Pulmonary arterial blood volume
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
Time
false
http://bhi.washington.edu/OPB#OPB01023
in
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
pulmonary_vein
false
End systolic pulmonary vein elastance
false
in
Left ventricular blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Blood pressure in the pulmonary artery
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
out
Pulmonary venous blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Thoracic cavity pressure
false
in
Blood flow through mitrial valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Pulmonary blood flow
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Mitrial valve fluid resistance
false
in
Pulmonary circulatory fluid resistance
false
false
pulmonary_vein_V_pu
Pulmonary venous blood volume
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
Time
false
http://bhi.washington.edu/OPB#OPB01023
in
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
right_ventricle
encapsulation
false
in
out
Right ventricular free wall end systolic elastance
false
in
out
Right ventricular free wall end diastolic pressure
false
in
Right ventricular free wall end diastolic pressure
false
in
Right ventricular free wall end systolic pressure
false
in
Blood pressure in the pulmonary artery
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Pericardial blood volume
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
out
right_ventricle_P_rv
Right ventricular blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
out
Pressure across the right ventricular free wall
false
in
Blood Pressure in the Vena Cava
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Blood flow through pulmonary valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Blood flow across tricuspid valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Pulmonary valve fluid resistance
false
in
Tricuspid valve fluid resistance
false
out
right_ventricle_V_rv
Right ventricle blood volume
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
out
out
Right ventricular free wall volume
Volume of spatial region
http://bhi.washington.edu/OPB#OPB_01061
false
in
Septal wall volume
Volume of spatial region
http://bhi.washington.edu/OPB#OPB_01061
false
in
Heart beat driver function
false
in
out
Right Ventricular Freewall Diastolic Exponential Parameter
Time
false
http://bhi.washington.edu/OPB#OPB01023
in
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
rvf_calculator
false
out
Right ventricular free wall end systolic elastance
false
out
Right ventricular free wall end diastolic pressure
false
out
Right ventricular free wall end diastolic pressure
false
out
Right ventricular free wall end systolic pressure
false
Right ventricular free wall volume at end diastolic pressure
false
Right ventricular free wall volume at zero pressure
false
in
Right ventricular free wall volume
Volume of spatial region
http://bhi.washington.edu/OPB#OPB_01061
false
out
Right Ventricular Freewall Diastolic Exponential Parameter
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
septum
false
in
Left ventricular free wall end systolic elastance
false
in
Right ventricular free wall end systolic elastance
false
Septal end systolic elastance
false
in
End diastolic pressure across the left ventricular wall
false
in
Right ventricular free wall end diastolic pressure
false
Septal wall end diastolic pressure
false
in
Left ventricular blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Right ventricular blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
Septal wall volume at end diastolic pressure
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
Septal wall volume at zero pressure
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
in
Volume of blood contained in left ventricle
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
in
Right ventricle blood volume
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
false
out
Septal wall volume
Volume of spatial region
http://bhi.washington.edu/OPB#OPB_01061
false
in
Heart beat driver function
false
in
Left Ventricular Freewall Diastolic Exponential Parameter
false
in
Right Ventricular Freewall Diastolic Exponential Parameter
false
Septal Wall Diastolic Exponential Parameter
false
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>
vena_cava
false
Vena cava end systolic elastance
false
in
Pressure in systemic arteries
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Right ventricular blood pressure
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
out
vena_cava_P_vc
Blood Pressure in the Vena Cava
Fluid Pressure
http://bhi.washington.edu/OPB#OPB_00509
false
in
Systemic circulatory blood flow
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Blood flow across tricuspid valve
Fluid Flow Rate
http://bhi.washington.edu/OPB#OPB_00299
false
in
Systemic circulatory fluid resistance
false
in
Tricuspid valve fluid resistance
false
Vena cava volume at zero pressure
false
vena_cava_V_vc
Vena cava blood volume
Fluid volume
http://bhi.washington.edu/OPB#OPB_00154
Time
false
http://bhi.washington.edu/OPB#OPB01023
in
<math xmlns="http://www.w3.org/1998/Math/MathML">
<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>
</math>