Left heart and systemic arteries pressure-flow model ported from Rideout (ACSL programs LH-PF). Also in MATLAB.
Further reading: Rideout's section 4.2, pages 78-90.
This model simulates the left heart using Pressure-Flow-Volume (or Pressure-Flow for short) modeling. This technique takes advantage of the analogy to RC or RLC circuits as shown here: Pressure-Flow-Volume Electrical Circuit Equivalent ------------------------------------------------------------------ F Flow (ml/s) I Current (Amper) P Pressure (mmHg or g/cm/s^2) V Voltage (Volt) Q Volume (ml) Q Charge (Coulomb or Amper*s) R Resistance (g/cm^4/s) R Resistance (Ohm or Volt/Amper) C Compliance (cm^4*s^2/g) C Capacitance (Farad or s/Ohm) L Inertance (g/cm^4) L Inductance (Henry or s*Ohm) Pressure-Flow components obey the same fundamental equations as their RLC circuit equivalents, i.e.: P = R * F V = R * I (Eq. 1 or Ohm's Law) P = L * F:t V = L * I:t (Eq. 2) P = Q / C V = Q / C (Eq. 3) F = Q:t I = Q:t (Eq. 4) Blood flows from the left atrium to the left ventricle via the mitral valve and continues to the aorta via the aortic valve. The load consists of systemic and capillary arteries. Atrial pressure (PAT) is assumed fixed at 6 mmHg as is the central venous pressure (PSV) at 3 mmHg. These are equivalent to input and output electric potentials (voltages) which drive the flow (current) through the left heart. Mitral and aortic valves are modeled by limiting the flow to be positive and less than an upper bound. There are three separate modes implemented in this model, corresponding to Rideout's ACSL programs: (1) LH-PF-1: Rectangular Actuator (2) LH-PF-2: Clipped Half-Sine (3) LH-PF-3: Skewed Half-Sine and Inertance LH-PF-1 and LH-PF-2 use a basic RC circuit; LH-PF-3 uses an RLC circuit with additional components. The actuating signal for all modes is a variable periodic left ventricular stiffness SLV (SLV = 1/CLV) which simulates a contracting and expanding left ventricle during the cardiac cycle. LH-PF-1 uses a rectangular SLV stiffness SLV waveform. LH-PF-2 uses a clipped half-sine. LH-PF-3 uses a clipped half-sine with a second harmonic term which adds some skew to the waveform. The figures show some results when running the JSim model in LH-PF-3. Figure 1 is the ventricular stiffness, SLV. Stiffness changes between a maximum, SLS during systole which lasts 0.3 sec, and a minimum, SLD during diastole which last 0.5 sec. The pulse is periodic with a period of 0.8 sec (corresponding to 75 beats/min heart pulse). Four heart beats are simulated for a total run time of 3.2 sec. The left ventricle volume, QLV expands during diastole and contracts during systole. Figure 2 shows the locus (path) of the left ventricular pressure, PLV, plotted versus volume, QLV. Values in this model were chosen so that the end-diastolic volume (EDV) is 120 ml and the end-systolic volume (ESV) is 48 ml. These volumes can be derived from Equation 3, P = Q / C, or Q = P / S and are typical in a healthy human adult. The ejection fraction (EDV - ESV) / EDV is 60% and stroke volume EDV - ESV is 72 ml. At a heart pulse of 0.8 sec, this corresponds to a cardiac output of 90 ml/sec (5.4 liter/min). The locus is traversed in counterclockwise direction. Note that it lies between the lines whose slopes are the diastolic (minimum) muscle stiffness SLD and the systolic (maximum) stiffness, SLS. Figure 3 shows various flows in the model. Note that FAT, the atrial flow rises then decays exponentially at diastole. This represents the diastolic inflow, which occurs when the walls of the left ventricle are relaxed and flow can pass through the mitral valve. As this chamber fills and expands, FAT decays. FLV, FAO and FSA have reverse activity, peaking and decaying during systole. FLV is the ventricular outflow and is limited by the aortic valve. FCA, capillary arteries flow, represents the systemic load and is much smaller and less pulsatile. Figure 4 shows various pressures in the model. PLV, the ventricular pressure increases during systole, closing the mitral valve, and with further increase opens the aortic valve. Aortic pressure PAO follows ventricular pressure PLV during systole, while load pressure PCA is much more rounded. During diastole, PAO and PCA nearly coincide. The slow exponential decay has a time constant TCA = RCA * CCA of 2.53 sec. Note that some oscillations are visible in PAO starting at the end of systole.
The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.
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Rideout VC. Mathematical computer modeling of physiological systems. Englewood Cliffs, NJ: Prentice Hall, 1991, 261 pp.
Please cite https://www.imagwiki.nibib.nih.gov/physiome in any publication for which this software is used and send one reprint to the address given below:
The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
Model development and archiving support at https://www.imagwiki.nibib.nih.gov/physiome provided by the following grants: NIH U01HL122199 Analyzing the Cardiac Power Grid, 09/15/2015 - 05/31/2020, NIH/NIBIB BE08407 Software Integration, JSim and SBW 6/1/09-5/31/13; NIH/NHLBI T15 HL88516-01 Modeling for Heart, Lung and Blood: From Cell to Organ, 4/1/07-3/31/11; NSF BES-0506477 Adaptive Multi-Scale Model Simulation, 8/15/05-7/31/08; NIH/NHLBI R01 HL073598 Core 3: 3D Imaging and Computer Modeling of the Respiratory Tract, 9/1/04-8/31/09; as well as prior support from NIH/NCRR P41 RR01243 Simulation Resource in Circulatory Mass Transport and Exchange, 12/1/1980-11/30/01 and NIH/NIBIB R01 EB001973 JSim: A Simulation Analysis Platform, 3/1/02-2/28/07.