2.1. Biomechanical model of LV passive dynamics

In this section, we will introduce LV passive dynamics in diastole. A human LV model from our previous studies are used here as shown in figure 1 with 53 548 nodes and 48 050 hexagonal elements. A rule-based approach is used to generate the layered myofibre structures within the myocardium, they are the fibre direction (f), the sheet direction (s) and the sheet-normal (n). In this work, the fibre angle α linearly rotates from −60° to 60° from endocardium to epicardium, and the sheet angle β linearly rotates from −45° to 45° in a similar way, and n = f × s. Details of the LV model reconstruction can be found in [7,8].

Visualization of the LV geometry. (f, s, n) are the fibre, sheet and sheet-normal axes, as described in the text, and (Wc, Wl, Wr) are coordinate axes that indicate the local circumferential, longitudinal and radial axes. The helix angle α is defined to be the angle between f and Wc in the plane spanned by Wc and Wl, and the sheet angle β is defined to be the angle between s and Wr in the plane spanned by Wl and Wr. The grey colour represents the epicardium and the green colour represents the endocardium.

The myocardium is described by a nearly incompressible orthotropic hyper-elastic material strain energy function (Ψ) developed in [12], namely the H-O law,

where a, b, af, bf, as, bs, afs, bfs are material parameters, the term (1/2)K (J − 1)2 accounts for the incompressibility of myocardium, and K is a constant bulk modulus (106 Pa). I1, I4i, I8fs (i = f, s) are the invariants along myofibre, sheet and sheet-normal directions, respectively,

in which C = FTF is the right Cauchy–Green deformation tensor and F is the deformation gradient. f0, s0 and n0 are the layered fibre structure in the reference configuration. In the current configuration, the fibre structure is defined as

The passive response of the LV dynamics in diastole is implemented and solved using the finite-element (FE) method in a general-purpose FE package ABAQUS (Simulia, Providence, RI, USA). The LV basal surface is fixed in the long-axial direction (Wl-axis) and the circumferential direction (Wc-axis), but allowing radial expansion, see figure 1. A linearly ramped pressure from 0 to 8 mm Hg is applied to the endocardial surface with 25 equal loading steps, and results are saved at each step. The LV cavity volume and principal strains at certain locations are chosen from the forward FE simulations, they are the maximum principal strain (ɛmax), which is related to myofibre stretch, and the minimum principal strain (ɛmin), which is related to wall thinning in diastole. In detail, to extract principal strains, 20 locations within the LV wall are randomly chosen using random function in Matlab [35], and then the maximum and minimum principal strains are spatially averaged at each loading step. Note that we only select 20 random positions once, and the same 20 positions are used for different simulations to extract strain data. The ventricular cavity volume is the volume enclosed by the endocardial surface. The scatter point in figure 2 shows the relationships between the pressure and the cavity volume, the mean maximum and minimum principal strains from one simulation in diastole. Published studies have found that exponential functions can characterize the nonlinear relationship between the pressure and the LV cavity volume very well [36]. For example, based on ex vivo human heart experiments, Klotz et al. [36] found that the relationship between the normalized volume (vn) and the loaded pressure (p) can be approximated with p=AnvnBn, in which An and Bn are coefficients, and both are almost invariant among subjects and species. Thus, in this study, we assume the relationships between the pressure and the LV cavity volume, the mean maximum and minimum principal strains also comply with the exponential function, as suggested by Klotz et al. [36], specifically

in which vn = (vv0)/v0 is the normalized volume with respect to the initial value v0, ε¯max and ε¯min are the mean maximum and minimum principal strains at chosen 20 positions. α0 and β0 can be least-square fitted to the pvn curve. α1 and β1 are derived from the pε¯max curve, and α2 and β2 are derived from the pε¯min curve. Because the minimum principal strain is negative, we take its absolute value in equation (2.4). Figure 2 shows the results from one simulation, the pvn, pε¯max and pε¯min are all fitted well with equation (2.4). Therefore, the output features from a forward ABAQUS simulation are reduced to three pairs of data for describing LV dynamics in diastole, rather than three different curves discretized with 75 data points.

A simple example of end-diastolic pressure–normalized volume and pressure–mean principal strain relationships. The discrete points are the original data from the forward simulation, the solid line is the curve fitting to the original data using the expressions in equation (2.4). (a) represents the curve fitting to the pressure and the normalized volume. (b) and (c) represent the curve fitting between the pressure and the mean maximum, minimum principal strains, respectively.

It has been shown that there is a strong correlation among the eight parameters in equation (2.1) [35], thus, it can be very challenging to uniquely determine all eight parameters by using only end-diastolic strains and volume. Following two recent studies from Noe et al. [26] and Davies et al. [27], the eight-dimensional parameter space is projected into a four-dimensional space,

where q = (q1, q2, q3, q4) ∈ [0.1, 5]4 are the reduced parameters, a0 = 0.22 kPa, b0 = 1.62, af0 = 2.43 kPa, bf0 = 1.83, as0 = 0.39 kPa, bs0 = 0.77, afs0 = 0.39 kPa, bfs0 = 1.70 are the empirical reference values for a healthy LV model [37]. The range of q is adopted from [26,27] which was derived from the population average values reported in [37].

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