Continuous flow left ventricular pump support and its effect on regional left ventricular wall stress: finite element analysis study

Abstract

Left ventricular assist device (LVAD) support unloads left ventricular (LV) pressure and volume and decreases wall stress. This study investigated the effect of systematic LVAD unloading on the 3-dimensional myocardial wall stress by employing finite element models containing layered fiber structure, active contractility, and passive stiffness. The HeartMate II^® (Thoratec, Inc., Pleasanton, CA) was used for LV unloading. The model geometries and hemodynamic conditions for baseline (BL) and LVAD support (LV_support) were acquired from the Penn State mock circulatory cardiac simulator. Myocardial wall stress of BL was compared with that of LV_support at 8,000, 9,000, 10,000 RPM, providing mean pump flow (Q _mean) of 2.6, 3.2, and 3.7 l/min, respectively. LVAD support was more effective at unloading during diastole as compared to systole. Approximately 40, 50, and 60 % of end-diastolic wall stress reduction were achieved at Q _mean of 2.6, 3.2, and 3.7 l/min, respectively, as compared to only a 10 % reduction of end-systolic wall stress at Q _mean of 3.7 l/min. In addition, there was a stress concentration during systole at the apex due to the cannulation and reduced boundary motion. This modeling study can be used to further understand optimal unloading, pump control, patient management, and cannula design.

Appendix

1.1 Constitutive equations for myocardium

1.1.1 Diastolic material properties

Diastolic material properties are represented by the strain energy function, W, to describe the myocardium with respect to the local muscle fiber direction as,

$$W = \frac{C}{2}\left\{ {\exp \left[ {b_{f} E_{11}^{2} + b_{t} (E_{22}^{2} + E_{33}^{2} + E_{23}^{2} + E_{32}^{2} ) + b_{fs} (E_{12}^{2} + E_{21}^{2} + E_{13}^{2} + E_{31}^{2} )} \right] - 1} \right\}$$

(1)

where the material constant C controls the myocardial stiffness, and material constants b _f , b _t , and b _fs govern the degree of anisotropy. E ₁₁ is fiber strain, E ₂₂ is cross-fiber strain, E ₃₃ is radial strain, E ₂₃ is shear strain in the transverse plane, and E ₁₂ and E ₁₃ are shear strain in the fiber-cross fiber and fiber-radial planes [34].

1.1.2 Systolic material properties

Systolic material properties are determined by defining the stress components referred to fiber coordinates. The systolic fiber stress is described as the sum of the passive stress components derived from the strain energy function W and an active fiber-direction component, T ₀ , that is a function of time, t, peak intracellular calcium concentration, Ca ₀ , sarcomere length, l, and the maximum isometric tension, T _max [34],

$$\tilde{S} = - pJ\tilde{C}^{ - 1} + \frac{\partial W}{{\partial \tilde{E}}} + T_{0} \{ t,Ca_{0} ,l,T_{\hbox{max} } \}$$

(2)

$\tilde{S}$ is the second Piola-Kirchoff stress tensor, p is a Lagrange multiplier introducing the incompressibility constraint, and the value was adopted from the bulk modulus of heart tissue [28], J is the Jacobian of the deformation gradient tensor $\tilde{F}$, $\tilde{C}$ is the right Cauchy-Green deformation tensor, and W is the strain energy function in Eq. (1). A time-varying elastance model at end-systole is given by

$$T_{0} = T_{\hbox{max} } \frac{{Ca_{0}^{2} }}{{Ca_{0}^{2} + ECa_{50}^{2} }}C_{t}$$

(3)

T _max is the maximum isometric tension achieved at the longest sarcomere length and maximum peak intracellular calcium concentration, (Ca ₀ )_max, and C _t is given by

$$C_{t} = \frac{1}{2}\left( {1 - \cos \omega } \right),\quad \omega = \pi \frac{{0.25 + t_{r} }}{{t_{r} }},\quad t_{r} = ml + b$$

(4)

where m and b are constants. The length-dependent calcium sensitivity is given by

$$ECa_{50} = \frac{{\left( {Ca} \right)_{\hbox{max} } }}{{\sqrt {\exp \{ B(l - l_{0} )\} - 1} }},\quad l = l_{R} \sqrt {2E_{ff} + 1}$$

(5)

where B is constant, l ₀ is the sarcomere length at which no active tension develops, and l _R is the stress-free sarcomere length. Finally, the Cauchy stress tensor used to calculate myocardial fiber stress is given by

$$\tilde{\sigma } = \frac{1}{J}\tilde{F}\tilde{S}\tilde{F}^{T}$$

(6)