Deriving the End-Systolic and End-Diastolic Pressure-Volume Relationships
Geometric assumptions. The left ventricle is approximated as a thick-walled sphere that undergoes centrosymmetric deformation, meaning it only experiences radial displacement during expansion and contraction. The myocardium is assumed to be incompressible, ensuring the tissue volume remains constant during these deformations. Geometrical variables, such as radius and wall thickness, are mathematically normalised using the inner radius and ventricular volume of the baseline reference geometry.
Mechanical work principle. The fundamental relationship between ventricular pressure and ventricular volume is established using the classical mechanics equation
where W represents the total elastic energy stored in the myocardium. This energy is calculated by integrating an energy density function over the spherical domain defined by the normalised wall thickness.
Deriving the EDPVR (passive mechanics). To characterise the relaxed, end-diastolic state, the model relies on an isotropic energy density function:
In this constitutive law, the mechanical parameters a and b represent the material stiffness of the cardiac tissue. Substituting this function into the mechanical work integral generates the EDPVR, isolating the passive pressure contributions of the myocardium.
Deriving the ESPVR (active mechanics). The contracted, end-systolic state combines passive tissue resistance with active force generation using an additive stress approach, through an active energy function:
where Ta is the maximum active stress and λ corresponds to sarcomere strain. Integrating this active contribution and adding it to the previously derived passive pressure equation yields the full ESPVR model.
Interactive Simulator
Adjust the myocardial properties (wall thickness, contractility, stiffness, fibrosis) and the hemodynamic load (preload, afterload, arterial compliance) to see the pressure-volume loop respond in real time, together with the resulting stroke volume, ejection fraction and blood pressure.