Computational Modeling Studies of the Roles of Left Ventricular Geometry, Afterload, and Muscle Contractility on Myocardial Strains in Heart Failure with Preserved Ejection Fraction

Abstract

Global longitudinal strain and circumferential strain are found to be reduced in HFpEF, which some have interpreted that the global left ventricular (LV) contractility is impaired. This finding is, however, contradicted by a preserved ejection fraction (EF) and confounded by changes in LV geometry and afterload resistance that may also affect the global strains. To reconcile these issues, we used a validated computational framework consisting of a finite element LV model to isolate the effects of HFpEF features in affecting systolic function metrics. Simulations were performed to quantify the effects on myocardial strains due to changes in LV geometry, active tension developed by the tissue, and afterload. We found that only a reduction in myocardial contractility and an increase in afterload can simultaneously reproduce the blood pressures, EF and strains measured in HFpEF patients. This finding suggests that it is likely that the myocardial contractility is reduced in HFpEF patients.

Appendix

Closed Loop Systemic Circulatory Model

The LV FE model was coupled to a closed loop lumped-parameter circulatory model that describes the systemic circulation (Fig. 1), which is similar to our previous work [27]. The modeling framework consists of five compartments of the systemic circulation namely, LV, LA, proximal artery, distal artery, and vein. The total mass of blood needs to be conserved in the circulatory model, which requires that the rate of volume change in each storage compartment of the circulatory system to the inflow and outflow rates by the following relations,

$$ \frac{d{V}_{LA}(t)}{dt}={q}_{ven}(t)-{q}_{mv}(t), $$

(1a)

$$ \frac{d{V}_{LV}(t)}{dt}={q}_{mv}(t)-{q}_{ao}(t), $$

(1b)

$$ \frac{d{V}_{a,p}(t)}{dt}={q}_{ao}(t)-{q}_{a,p}(t), $$

(1c)

$$ \frac{d{V}_{a,d}(t)}{dt}={q}_{a,p}(t)-{q}_{a,d}(t), $$

(1d)

$$ \frac{d{V}_{ven}(t)}{dt}={q}_{a,d}(t)-{q}_{ven}(t), $$

(1e)

where V_LA, V_LV, V_{a, p}, V_{a, d}, and V_ven are volumes of LV, LA, proximal artery, distal artery, and vein, respectively, and q_ven, q_mv, q_ao, q_{a, p}, and q_{a, d} are flow rates at different segments. Flowrate at different segments of the circulatory model depends on their resistance to flow (R_ao, R_{a, p}, R_{a, d}, R_ven, and R_mv) and the pressure difference between the connecting storage compartments (i.e., pressure gradient). The flow rates are given by

$$ {q}_{ao}(t)=\left\{\begin{array}{cc}\frac{P_{LV}(t)-{P}_{a,p}(t)}{R_{ao}}&\ when,{P}_{LV}(t)\ge {P}_{a,p}(t)\kern0.5em \\ {}0& when,{P}_{LV}(t)<{P}_{a,p}(t)\end{array}\right., $$

(2a)

$$ {q}_{a,p}(t)=\frac{P_{a,p}(t)-{P}_{a,d}(t)}{R_{a,p}}, $$

(2b)

$$ {q}_{a,d}(t)=\frac{P_{a,d}(t)-{P}_{ven}(t)}{R_{a,d}}, $$

(2c)

$$ \kern0.50em {q}_{ven}(t)=\frac{P_{ven}(t)-{P}_{LA}(t)}{R_{ven}}, $$

(2d)

$$ {q}_{mv}(t)=\left\{\begin{array}{cc}\frac{P_{LA}(t)-{P}_{LV}(t)}{R_{mv}}&\ when,{P}_{LA}(t)\ge {P}_{LV}(t)\kern0.5em \\ {}0& when,{P}_{LA}(t)<{P}_{LV}(t)\end{array}.\right. $$

(2e)

Pressure in each storage compartment is a function of its volume. A simplified pressure–volume relationship

$$ {P}_{a,p}(t)=\frac{V_{a,p}(t)-{V}_{ap,0}}{C_{a,p}}, $$

(3a)

$$ {P}_{a,d}(t)=\frac{V_{a,d}(t)-{V}_{ad,0}}{C_{a,d}}, $$

(3b)

$$ {P}_{ven}(t)=\frac{V_{ven}(t)-{V}_{ven,0}}{C_{ven}}, $$

(3c)

was prescribed for the proximal artery, distal artery, and veins, where V_{ap, 0}, V_{ad, 0}, and V_{ven, 0} are constant resting volumes of the proximal artery, distal artery, and veins. C_{a, p}, C_{a, d}, and C_ven are the total compliance of the proximal artery, distal artery, and venous system. On the other hand, pressure in the left atrium P_LA(t) was prescribed to be a function of its volume V_LA(t) by the following equations that describe its contraction using a time-varying elastance function [67]

$$ {P}_{LA}(t)=e(t){P}_{es, LA}\left({V}_{LA}\left(\mathrm{t}\right)\right)+\left(1-e(t)\right)\ {P}_{ed, LA}\left({V}_{\mathrm{LA}}\left(\mathrm{t}\right)\right), $$

(4)

where

$$ {P}_{es, LA}\left({V}_{LA}\left(\mathrm{t}\right)\right)={E}_{es, LA}\left({V}_{LA}\left(\mathrm{t}\right)-{V}_{0, LA}\right), $$

(5a)

$$ {P}_{ed, LA}\left({V}_{LA}\left(\mathrm{t}\right)\right)={A}_{LA}\ \left({e}^{B_{LA}\left({V}_{LA}\left(\mathrm{t}\right)-{V}_{0, LA}\right)}-1\right), $$

(5b)

and,

$$ e(t)=\left\{\begin{array}{cc}\frac{1}{2}\left(\sin \left[\left(\frac{\pi }{t_{max}}\right)t\hbox{--} \frac{\pi }{2}\right]+1\right);& \kern0.5em 0<t\le 3/2\ {t}_{max}\\ {}\frac{1}{2}\ {e}^{-\left(t-3/2{t}_{max}\right)/{\tau}_{LA}};& \kern0.5em t>3/2\ {t}_{max}\end{array}.\right. $$

(5c)

In Eqs. (5a–b), E_{es, LA} is the end-systolic elastance of the left atrium, V_{0, LA} is the volume axis intercept of the end-systolic pressure–volume relationship (ESPVR), and both A_LA and B_LA are parameters of the end-diastolic pressure–volume relationship (EDPVR) of the left atrium. The driving function e(t) is given in Eq. (5c) in which t_max is the point of maximal chamber elastance and τ is the time constant of relaxation. The values of E_{es, LA}, V_{0, LA}, A_LA, B_LA, t_max, and τ_LA are listed in Table 3.

Finally, pressure in the LV depends on its corresponding volume through nonclosed form function

$$ {P}_{LV}(t)={f}^{LV}\left({V}_{LV}(t)\right), $$

(6)

The functional relationship between pressure and volume in the LV was obtained using the FE method as described in the next section. Parameter values associated with the closed loop circulatory model are tabulated in Table 4.

Table 3 Fixed parameters of LA time varying elastance model for all cases

Table 4 Fixed parameter values of the circulatory model for all simulation cases

Finite Element Formulation of the LV

The weak form associated with finite element formulation of the LV was derived based on the minimization of the following Lagrangian functional [28, 37],

$$ \mathcal{L}\left(\boldsymbol{u},p,{P}_{\mathrm{cav}},{\boldsymbol{c}}_1,{\boldsymbol{c}}_2\right)={\int}_{\Omega_0}W\left(\boldsymbol{u}\right) dV-{\int}_{\Omega_0}p\left(J-1\right) dV-{P}_{\mathrm{cav}}\left({V}_{\mathrm{cav}}\left(\boldsymbol{u}\right)-V\right)-{\boldsymbol{c}}_1\cdotp {\int}_{\Omega_0}\boldsymbol{u}\ dV-{\boldsymbol{c}}_2\cdotp {\int}_{\Omega_0}\boldsymbol{X}\times \boldsymbol{u}\ dV, $$

(7)

where, u is the displacement field, P_cav is the Lagrange multiplier to constrain the LV cavity volume V_cav(u) to a prescribed value V [68], p is a Lagrange multiplier to enforce incompressibility of the tissue (i.e., Jacobian of the deformation gradient tensor J = 1), and both c₁ and c₂ are Lagrange multipliers to constrain rigid body translation (i.e., zero mean translation) and rotation (i.e., zero mean rotation) [[69]]. The LV cavity volume V_cav is a function of the displacement u and is defined by

$$ {V}_{\mathrm{cav}}\left(\boldsymbol{u}\right)=\underset{\Omega_{inner}}{\int } dv=-\frac{1}{3}\underset{\Gamma_{inner}}{\int}\boldsymbol{x}.\boldsymbol{n}\ da, $$

(8)

where Ω_inner is the volume enclosed by the inner surface Γ_inner and the basal surface at z = 0, and n is the outward unit normal vector.

Pressure–volume relationship of the LV required in the lumped parameter circulatory model (i.e., Eqs. (6)) was defined by the solution obtained from minimizing the functional [27]. Taking the first variation of the functional in Eq. (7) leads to the following expression:

$$ \delta \mathcal{L}\left(\boldsymbol{u},p,{P}_{\mathrm{cav}},{\boldsymbol{c}}_1,{\boldsymbol{c}}_2\right)={\int}_{\Omega_0}\left(\boldsymbol{P}-p{\boldsymbol{F}}^{-\boldsymbol{T}}\right):\nabla \delta \boldsymbol{u}\ dV-{\int}_{\Omega_0}\delta p\left(J-1\right) dV-{P}_{\mathrm{cav}}{\int}_{\Omega_0} cof\left(\boldsymbol{F}\right):\nabla \delta \boldsymbol{u}\ dV-\delta {P}_{\mathrm{cav}}\left({V}_{\mathrm{cav}}\left(\boldsymbol{u}\right)-V\right)-{\delta \boldsymbol{c}}_1\cdotp {\int}_{\Omega_0}\boldsymbol{u}\ dV-\delta {\boldsymbol{c}}_2\cdotp {\int}_{\Omega_0}\boldsymbol{X}\times \boldsymbol{u}\ dV-{\boldsymbol{c}}_1\cdotp {\int}_{\Omega_0}\delta \boldsymbol{u}\ dV-{\boldsymbol{c}}_2\cdotp {\int}_{\Omega_0}\boldsymbol{X}\times \delta \boldsymbol{u}\ dV. $$

(9)

In Eq. (9), P is the first Piola Kirchhoff stress tensor, F is the deformation gradient tensor, δu, δp, δP_cav, δc₁, δc₂ are the variation of the displacement field, Lagrange multipliers for enforcing incompressibility and volume constraint, zero mean translation and rotation, respectively. The Euler-Lagrange problem then becomes finding u ∈ H¹(Ω₀), p ∈ L²(Ω₀), P_cav ∈ ℝ, c₁ ∈ ℝ³, c₂ ∈ ℝ³ that satisfies

$$ \delta \mathcal{L}\left(\boldsymbol{u},p,{P}_{\mathrm{cav}},{\boldsymbol{c}}_1,{\boldsymbol{c}}_2\right)=0 $$

(10)

and u(x, y, 0).n|_base = 0 (for constraining the basal deformation to be in-plane) ∀δu(Ω₀), δp ∈ L²(Ω₀), δP_cav ∈ ℝ, δc₁ ∈ ℝ³, δc₂ ∈ ℝ³.

Constitutive Law of the LV

An active stress formulation was used to describe the LV’s mechanical behavior in the cardiac cycle. In this formulation, the stress tensor P can be decomposed additively into a passive component P_p and an active component P_a (i.e., P = P_a + P_p). The passive stress tensor was defined by P_p = dW/dF, where W is a strain energy function of a Fung-type transversely isotropic hyperelastic material [30] given by

$$ W=\frac{1}{2}C\left({e}^Q-1\right), $$

(11a)

where,

$$ Q={b}_{ff}{E}_{ff}^2+{b}_{xx}\left({E}_{ss}^2+{E}_{nn}^2+{E}_{sn}^2+{E}_{ns}^2\right)+{b}_{fx}\left({E}_{fn}^2+{E}_{nf}^2+{E}_{fs}^2+{E}_{sf}^2\right). $$

(11b)

In Eq. (11), E_ij with (i, j) ∈ (f, s, n) are components of the Green–Lagrange strain tensor E with f, s, n denoting the myocardial fiber, sheet, and sheet normal directions, respectively. Material parameters of the passive constitutive model are denoted by C, b_ff, b_xx, and b_fx.

The active stress P_a was calculated along the local fiber direction using a modified time varying elastance model,

$$ {\boldsymbol{P}}_{\mathrm{LV},\mathrm{a}}={T}_{ref}\frac{Ca_0^2}{Ca_0^2+{ECa}_{50}^2}{C}_t\ {\mathbf{e}}_f\otimes {\mathbf{e}}_{f_0} $$

(12)

In the above equation, e_f and ${\mathbf{e}}_{f_0}$ are, respectively, the local vectors defining the muscle fiber direction in the current and reference configuration, T_ref is the reference tension and Ca₀ denotes the peak intracellular calcium concentration. The length dependent calcium sensitivity ECa₅₀ and the variable C_t are given by [29]

$$ {ECa}_{50}=\frac{{\left({Ca}_0\right)}_{max}}{\sqrt{\exp \left(B\left(l-{l}_0\right)\right)-1}}, $$

(13)

$$ C(t)=\left\{\begin{array}{c}\frac{1}{2}\left(1-\mathit{\cos}\left(\pi \frac{t}{t_0}\right)\right)\kern6.5em t<{t}_t\\ {}\frac{1}{2}\left(1-\mathit{\cos}\left(\pi \frac{t_t}{t_0}\right)\right)\mathit{\exp}\left(\frac{t-{t}_t}{\tau}\right)\kern1em t\ge {t}_t\end{array}\right. $$

(14)

In Eq. (13), B is a constant, (Ca₀)_max is the maximum peak intracellular calcium concentration and l₀ is the sarcomere length at which no active tension develops. In Eq. (14), t₀ is the time taken to reach the peak tension, t_t is the time at which isovolumic relaxation of LV starts and τ is the time constant of the isovolumic relaxation. Parameter values associated with the LV model are tabulated in Table 5.

Table 5 Fixed parameter values of the LV FE model

Parameters for each simulation cases are tabulated in Table 6.

Table 6 Parameters for simulation cases