Assessment of wall stresses and mechanical heart power in the left ventricle: Finite element modeling versus Laplace analysis

Abstract Introduction Stenotic aortic valve disease (AS) causes pressure overload of the left ventricle (LV) that may trigger adverse remodeling and precipitate progression towards heart failure (HF). As myocardial energetics can be impaired during AS, LV wall stresses and biomechanical power provide a complementary view of LV performance that may aide in better assessing the state of disease. Objectives Using a high‐resolution electro‐mechanical (EM) in silico model of the LV as a reference, we evaluated clinically feasible Laplace‐based methods for assessing global LV wall stresses and biomechanical power. Methods We used N = 4 in silico finite element (FE) EM models of LV and aorta of patients suffering from AS. All models were personalized with clinical data under pretreatment conditions. Left ventricle wall stresses and biomechanical power were computed accurately from FE kinematic data and compared with Laplace‐based estimation methods, which were applied to the same FE model data. Results and Conclusion Laplace estimates of LV wall stress are able to provide a rough approximation of global mean stress in the circumferential‐longitudinal plane of the LV. However, according to FE results, spatial heterogeneity of stresses in the LV wall is significant, leading to major discrepancies between local stresses and global mean stress. Assessment of mechanical power with Laplace methods is feasible, but these are inferior in accuracy compared with FE models. The accurate assessment of stress and power density distribution in the LV wall is only feasible based on patient‐specific FE modeling.

SUPPLEMENTARY MATERIAL

Derivation of Laplace's law
To derive the law of Laplace we consider a spherical shell of inner radius, r and thin walls of a given thickness, h. Inflating the shell by applying a pressure, p, within the shell's cavity, induces deformation which causes the buildup of stresses within the wall. If we consider one half of the sphere, the total force Fp acting on the inner surface must be balanced with the total force acting over the cut surface (see Fig. 1.A). Due to spherical symmetry, the circumferential stresses σ circ (ρ) at any radius r ≤ ρ ≤ R must be the same and the shear stress is zero. Integrating circumferential stresses over the cut surface yields the total force balancing the force due to the applied pressure. That is, we have p r 2 π ! = (R 2 − r 2 ) σ circ π. (1) Assuming that h r, radial stresses are small compared to circumferential stresses, σrr σ circ , and the total stress tensor is approximated by with respect to the spherical coordinate system and the projection matrix P = (er, eϕ, e θ ) , see Fig. 1.C. Note that σ in a spherical shell differs from a stress tensor in the LV in various ways. Unlike in the LV, stresses in circumferential and meridonial/longitudinal direction are equal whereas in the LV longitudinal stresses tend to be larger than circumferential stresses. Further, the assumption r h is not justified, rather r ≈ h holds.
Thus radial stresses in the LV are non-negligible, that is, σrr is at an order of magnitude comparable to σ circ .
x y z e r e θ e ϕ θ ϕ r 0 (1.C) FIGURE 1 Balance of forces in thin-walled spherical shell models, spherical coordinate system and displacement boundary conditions.

Laplace's law for a thick-walled sphere
From Eq. (1) the circumferential stress in a thick-walled sphere is found as which we denote as σ L,H .

Laplace's law for a thin-walled sphere
Using assumption (A3), i.e. h/r 1, we have 1 + h 2 r ≈ 1 which yields the simple law of Laplace for a thin-walled sphere given by which we denote as σ L,h .
which we denote as σ L,V .

Computation of power and work
The approximations σ L,h , σ L,H and σ L,V for the circumferential stress σ circ can be used to derive an estimator for the internal power and internal work For this sake, we consider Eq. (6) and the simplified representation of the total stress tensor Eq. 2. In Eq. (6) an approximation of the strain rateε is required. Rewriting the strain tensor ε in spherical coordinates, as done for the stress tensor σ, we obtain where P is the projection matrix introduced in Sec. 1.1. Similarly, the strain rateε is expressed aṡ Using Eq.
(2), an approximation of the internal power density, p int can be derived as (σ :ε) ≈ σ circ (εϕϕ +ε θθ ). Due to the assumption of symmetry (A2), strains in circumferential direction do not vary with space, i,e. εϕϕ = ε θθ = ε circ , and the approximation for the internal power density simplifies to An approximation of the circumferential strain ε circ can be found based the Cauchy strain and considerations illustrated in Fig. 2. Accordingly, for a given radius r circumferential strain can be approximated as and for the circumferential strain rate we obtainε circ ≈ṙ r0 .
To approximate the circumferential strain rateε circ (u, t) of a spherical shell of thickness h = R − r, we take the arithmetic mean of the strain rate at inner radius r and outer radius R, that isε where r0 is the initial inner radius and R0 is the initial outer radius of the spherical shell at its stress free configuration, i.e. p = 0.
Using Eqs. (8) and (9), the internal power can be estimated by where Vmyo(t) is the volume of the shell's wall at time t. Using an estimate for σ circ we obtain with ∈ {h, H, V} and by integrating over time, we get an estimate for the internal work

Model fitting
To delineate anisotropy from pure geometry effects, passive inflation experiments were also performed with LV models using the Demiray model and passive mechanical behavior was compared to the spherical shell models Sph 5 , Sph 25 and Sph 150 . In these cases, parameters were set to b = 7 and a was chosen in a patient-specific manner to obtain the same volume at maximum inflation pressure as with the Guccione model. Note that in none of the simulations of a full cardiac cycle the Demiray model was considered as the resulting kinematics was in stark contrast to the clinical data.

Analysis of LV inflation experiments
To evaluate the influence of violating the assumption on the geometry (A2), passive inflation experiments were performed with LV models and the isotropic material due to Demiray, following the same protocol as applied to the spherical shell models Sph 5 , Sph 25 and Sph 150 . The parameter a in the Demiray model was set to 0.45, 0.63, 0.41 and 0.38 kPa for the models LV A , LV B , LV C and LV D , respectively. The Laplace-based stress estimates σ L,h , σ L,H and σ L,V were compared to the mean stresses obtained from the FE solution. Stresses were evaluated with respect to an ellipsoidal coordinate system to facilitate a comparison with stresses computed in the spherical shell models Sph 5 , Sph 25 and Sph 150 where spherical coordinates were used for stress analysis. The ellipsoidal coordinate system for the LV models was constructed by assigning fiber and sheet orientations using a rule-based method with a constant fiber angle of 0 • . Stress components σrr(x), σϕϕ(x) and σ θθ (x) were averaged yielding σrr, σϕϕ and σ θθ , respectively.
Note that all models except Sph 5 showed marked spatial stress variations. Thus, the reported mean stressesσ may deviate considerably from the true local stresses σ(x). Laplace-based estimations of power P int, and work W int, , were compared to those obtained by FE simulation, P int and W int and to external hydrodynamic power and work in the LV cavity, P ext and W ext .

Verification of the FE model
Similarly, with increasing h the accuracy of the thick-walled Laplace estimate W int,H performed better than the simpler thin-walled Laplace estimate W int,h . As expected on grounds of conservation of energy, the agreement between biomechanical work W int and hemodynamic work W ext was essentially perfect with differences < 2% for all models.

Passive inflation of LV models
The LV models LV A -LV D were inflated following the loading protocol in Fig. 3.A. Passive material behavior was represented compliant with (A1) by the isotropic Demiray model. The temporal evolution of FE-and Laplace-based stresses, power and work are shown in Fig. 3.B for model LV D . Minor quantitative differences to other models LV A -LV C were observed, but qualitatively the overall behavior was identical. Stresses at p = 4 kPa and the amount of work incurred during inflation up to this pressure are summarized in Tabs

FIGURE 5
Differences between mechanical versus hydrodynamic power, ∆P = P int − P ext , during IVC and early ejection were very minor (dark yellow area). A slightly more pronounced ∆P is witnessed during late ejection and IVR (gray area).