Constitutive discovery in the living human heart

Figures

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  Fig. 1. Computational workflow of the invariant-based CANN embedded within a finite element (FE) solver. It provides the deformation gradient F at each integration point from which the third and eight isochoric invariants are evaluated and passed through the CANN to obtain the strain-energy function ψ and the stress measure P. These quantities are used to assemble the global FE system, solve for the displacement increment, and update the configuration iteratively until convergence of both, the load increment and the loss in Eq. (12).

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  Fig. 2. From left to right: Personalized fiber, sheet, and normal directions in the representative left ventricle.

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  Fig. 3. Displacement error between simulated and MRI-derived ventricular deformation at the end of the diastole. Corresponding prediction errors are displayed for the epicardial nodes used for the model discovery. Displayed left ventricular geometry is simulated with models and parameters specified in Tables 1 and 2.

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  Fig. 4. Stress profiles across human left ventricle at the end of the diastole. Von Misses stress generated by the left ventricle is simulated with models and parameters specified in Tables 1 and 2.

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  Fig. 5. Comparison of constitutive responses by the discovered model under ex vivo loading protocols (six simple shear and five biaxial tests). Colored regions indicate the contributions of the active constitutive terms and symbols denote the experimental data (Sommer et al. 2015).

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  Fig. 6. Comparison of different constitutive model responses under ex vivo loading protocols (one simple shear and one biaxial tests). Colored regions show the contributions of the active constitutive invariants and dots indicate experimental data (Sommer et al. 2015).