Physical model of end-diastolic and end-systolic pressure-volume relationships of a heart

Figures

• 

  FIGURE 1. Illustration of the PV loop, ESPVR, and EDPVR of the LV. The EDPVR and ESPVR are highlighted in red and the PV loop is in blue. The arrows on the PV loop correspond to the direction of the LV beating cycle. Curve 3-4-1 corresponds to the diastolic phase. At point 1, the ventricular volume reaches its maximum value as the blood fills into the LV, which is called the EDV. Similarly, curve 1-2-3 corresponds to the systolic phase. At point 3, the ventricular volume reaches its minimum value during LV contraction, which is referred to as ESV. The difference between EDV and ESV is the SV, indicating the amount of blood pumped by the LV per cardiac cycle. At different filling pressure and contractility, points 1 and 3 move on a single curve called the EDPVR and ESPVR, respectively. (EDPVR, end-diastolic pressure-volume relationship; ESPVR: end-systolic pressure-volume relationship; EDV, end-diastolic volume; ESV, end-systolic volume; SV, stroke volume).

• 

  FIGURE 2. Schematic diagram of the geometry and its deformation in the physics-based model. A thick-walled sphere is used to mimic the LV, whose cross section through the sphere center is shown. The reference geometry Ω0 with inner radius R endo and outer radius R epi is given on the left side. The deformed geometry Ω in the diastolic state with inner radius r endo and outer radius r epi is shown on the right. For any point X in the reference geometry Ω0, the corresponding position on the deformed geometry Ω is x, with radial coordinate R and r, respectively. Centrosymmetric deformation is assumed for the model. Endo, endocardium; epi, epicardium.

• 

  FIGURE 3. Comparison between the exponential model (Eq. 1), the Klotz curve (Eq. 2) and the physics-based model (Eq. 19) for the EDPVR. (A–C) The fitted curves to the full experimental dataset and the corresponding residuals. MSEs: exponential model, 2.82; Klotz curve, 3.19; physics-based model, 3.08. (B–D) The fitted curves to the sub-dataset and the corresponding residuals, with pressure no more than 20 mmHg. MSEs: exponential model, 2.48; Klotz curve, 2.69; physics-based model, 2.46. Model parameters for subfigures (A–C) and (B–D) are listed in Table 1.

• 

  FIGURE 4. The physics-based model predicts the effect of varying mechanical properties on the EDPVR with in silico validation. (A) The EDPVR curves predicted by the physics-based model with different parameter sets. (B) In silico validation of the physics-based model. Simulation results agree with the physics-based model very well for each set of parameters. Parameters used in the physics-based model: Klotz 2007: a = 1.15 kPa, b = 3.82; Dokos 2002: a = 2.52 kPa, b = 6.79; Demiray 1972: a = 1.00 kPa, b = 6.5; Marx 2022: a = 1.98 kPa, b = 6.19. For all curves: Δ = 0.27.

• 

  FIGURE 5. The physics-based model predicts the effect of varying myocardial thickness on the EDPVR. (A) The EDPVR curve moves upward with the increase of thickness Δ, while a and b remain constant. (B) Nonlinear relationship between the ventricular pressure and the thickness, with constant normalized volume V/V 0 = 2.0. Parameters used in the physics-based model: a = 1.15 kPa and b = 3.82.

• 

  FIGURE 6. The intrinsic structure and in silico validation of the physics-based ESPVR model. (A) The pressure in the ESPVR has two contributions, one from the active contraction and the other one from the passive resistance of the tissue. The overall level of pressure is mainly determined by the active stress. The shape, especially at the lower ventricular volume region from 0.5 to 0.7, is mostly influenced by passive resistance. (B) The theoretical ESPVR curve fits very well with the simulation results. The parameters for both the theoretical and simulation are Δ = 0.27, a = 1.15 kPa, b = 3.82, and T a = 76.9 kPa.

• 

  FIGURE 7. Comparison of the physics-based model and experimental data for the ESPVR. The physics-based ESPVR model is shown in blue. The experimental data of a pig heart, shown as orange crosses, is from Moriz et al. (2021). The volume of the original data is normalized, assuming stress-free volume V 0 = 50 mL. Parameters used in the physics-based model: Δ = 0.27, a = 2.52 kPa, b = 6.79, and T a = 81.89 kPa.

• 

  FIGURE 8. Effect of wall thickness on the EDPVR and ESPVR predicted by the physics-based model. Δ0 = 0.27 is the LV wall thickness of a healthy human heart and is therefore considered here as a reference case. The change in wall thickness has a significant effect on the ESPVR, but a relatively small effect on the EDPVR. Parameters used in the physics-based model: a =1.15 kPa, b =3.82, and T a =76.9 kPa.