Heterogeneous growth-induced prestrain in the heart

Abstract

Even when entirely unloaded, biological structures are not stress-free, as shown by Y.C. Fung׳s seminal opening angle experiment on arteries and the left ventricle. As a result of this prestrain, subject-specific geometries extracted from medical imaging do not represent an unloaded reference configuration necessary for mechanical analysis, even if the structure is externally unloaded. Here we propose a new computational method to create physiological residual stress fields in subject-specific left ventricular geometries using the continuum theory of fictitious configurations combined with a fixed-point iteration. We also reproduced the opening angle experiment on four swine models, to characterize the range of normal opening angle values. The proposed method generates residual stress fields which can reliably reproduce the range of opening angles between 8.7±1.8 and 16.6±13.7 as measured experimentally. We demonstrate that including the effects of prestrain reduces the left ventricular stiffness by up to 40%, thus facilitating the ventricular filling, which has a significant impact on cardiac function. This method can improve the fidelity of subject-specific models to improve our understanding of cardiac diseases and to optimize treatment options.

Keywords: Finite element method; Finite strain; Opening angle; Patient-specific modeling; Prestrain; Residual stress.

Figure 1

Kinematics of prestrained biological systems. Prestrain maps the stress-free reference configuration onto the residually stressed but mechanically unloaded configuration through the prestrain tensor $\underset{‗}{F^p}$. Mechanical loading maps the residually stressed but mechanically unloaded configuration into the mechanically loaded configuration through the deformation gradient $\underset{‗}{F}$. The total elastic deformation $\underset{‗}{F^e}$ is a result of prestrain $\underset{‗}{F^p}$ and mechanical deformation $\underset{‗}{F}$. Only the mechanical deformation $\underset{‗}{F}=\nabla \underset{\_}{\phi }$ is the gradient of a vector field, while the prestrain tensor $\underset{‗}{F^p}$ and the elastic tensor $\underset{‗}{F^e}$ are incompatible. Only the elastic tensor $\underset{‗}{F^e}$ generates stress.

Figure 2

Kinematics of growth-induced, prestrained biological systems. Growth turns the initial, stress-free and compatible configuration into the grown, stress-free but incompatible configuration. Prestrain maps the grown, stress-free configuration onto the residually stressed but mechanically unloaded configuration through the prestrain tensor $\underset{‗}{F^p}$. Mechanical loading maps the residually stressed but mechanically unloaded configuration into the mechanically loaded configuration through the deformation gradient ${\underset{‗}{F}}^{\prime }$. The total elastic deformation $\underset{‗}{F^e}$ is a result of prestrain $\underset{‗}{F^p}$ and mechanical deformation ${\underset{‗}{F}}^{\prime }$. The full deformation $\underset{‗}{F}$ is a result of growth $\underset{‗}{F^g}$, prestrain $\underset{‗}{F^p}$, and mechanical deformation ${\underset{‗}{F}}^{\prime }$. Only the full deformation $\underset{‗}{F}=\nabla \underset{\_}{\phi }$ and mechanically-induced deformation ${\underset{‗}{F}}^{\prime }=\nabla {\underset{\_}{\phi }}^{\prime }$ are gradients of a vector field, while the growth tensor $\underset{‗}{F^g}$, the prestrain tensor $\underset{‗}{F^p}$, and the elastic tensor $\underset{‗}{F^e}$ are incompatible. Note that Figure 2 becomes identical to Figure 1 if $\underset{‗}{F^g}=\underset{‗}{1}$.

Figure 3

Opening angle experiment. Isolated heart slice before (left) and after (right) radial cutting in the middle of the left ventricular free wall.

Figure 4

Protocol to generate heterogeneous growth-induced residual stress in a patient specific left ventricular geometry. The ventricle is loaded to a ventricular pressure of P̄, (a)–(b), then allowed to grow for a duration t̄, (b)–(c), and then unloaded, (c)–(d). The strain-driven growth drives the growing configuration into a state with a more homogeneous myofiber stress and reduces regional stress variations (c). The final configuration is unloaded, but not stress free, with the sub-endocardial region in compression, and the sub-epicardial region in tension (d).

Figure 5

Fixed-point iteration method to compute the prestrained, mechanically unloaded reference configuration (last column, top row) so that the prestrained, mechanically loaded configuration (last column, bottom row) matches the in vivo geometry extracted from magnetic resonance images (first column, top row) but contains auto-balanced residual stresses.

Figure 6

Prestrained configurations for varying prestrain levels. Prestrain is generated by heterogeneous growth. The reference configurations are computed using a fixed-point iteration, so that the prestrained configurations (bottom row) match the in vivo geometry extracted from magnetic resonance images (top row).

Figure 7

Virtual opening angle experiment. The prestrained configuration is sliced and then cut. The slice naturally springs open, similar to the physical experiment illustrated in Figure 3. Cutting the slice relieves some, but not all residual stress. Colors represent residual fiber stress, in kPa.

Figure 8

Opening angles for varying prestrain levels. The green curve summarizes the computationally simulated opening angles for a pressure of P̄ = 1 mmHg. The black line and gray box mark the experimentally measured opening angle of 13±5.3° for the equatorial slice. Computationally simulated opening angles lie within the range of experimentally measured opening angles.

Figure 9

Passive pressure-volume response of the left ventricle for varying prestrain levels. Increasing the prestrain level increases the ventricular compliance: for an opening angle of 17.1°, i.e., close to the average experimental value, the ventricular pressure after passive filling corresponding to a 50% ejection fraction is reduced by 25.8% compared to the prestrain-free case.