## 6 Applications to Cardiovascular Medicine
### The Heart
#### 6.2.1 Properties of the Healthy and Infarcted Ventricular Wall
Ventricular pressure-volume relation was assumed to follow the normalized Klotz LV pressure-volume curve, and the PC MRI-derived diastole-to-systole flow ratio was correlated to the pressure pulse amplitude. The authors proposed a data-based method for imposing boundary conditions through the use of Lagrange multipliers and the minimization of energy potentials. The endocardial boundary condition was defined in terms of volume change, while the basal plane and the epicardial node boundary conditions were defined in terms of a virtual force proportional to their correspondent displacements. A Shamanskii-Newton Raphson procedure was used to resolve the material properties and boundary condition multipliers. Parameter fitting was solved in two steps: first, the passive material parameters were solved by minimizing a displacement-based error function from tissue tagging data between early- and end-of-diastole; second, the active components were fitted by minimizing an error function defined as a weighted average of nodal displacement error and pressure-volume curve error. The authors highlight the potential bias introduced by hard displacement restrictions as boundary conditions, which prevents the reproduction of naturally occurring torsional modes of deformation. They also highlight that the direct imposition of noise MRI-derived displacement as a boundary condition can introduce computational issues associated with continuity and solution smoothness, concluding that the proposed weak formulation of boundary conditions is advantageous. [PERSON] et al. performed an inverse analysis of biventricular models with a microstructural material model [108]. Steady-state free precession (SSFP) cine MRI was used to build anatomical models, and to calculate the diastole-to-systole volume change of both ventricles from five healthy adult volunteers. A purely passive Holzapfel-Ogden material model was imposed with fiber orientation following the Laplace-Dirichlet rule-based algorithm by [PERSON] et al. [55]. Early diastole was assumed as the stress-free reference configuration. A normalized Klotz pressure-volume curve was set as the optimization target for a genetic algorithm. The authors introduced empiric constraints on the constitutive equation for the maximum absolute and relative values of shear-related terms to reduce the sampling space. In addition, they carried out a sensitivity analysis of the results on the assumed parameters. They concluded that variations within the normal ranges of interventricular pressure and fiber orientation did not produce significant changes in material parameters estimations. [PERSON] et al. explored potential differences in LV stiffness among heart failure patients with preserved and reduced ejection fractions [248]. Cine MRI and catheter pressure measurements were collected from 8, 11, and 5 individuals with reduced, preserved, and normal ejection fractions, respectively. Anatomical models of the LV were customized for all cine MRI images with an interactive guide-point modeling tool and volumes were matched to pressure measurements. The diastasis state (immediately after rapid filling) was assumed as the reference stress-free configuration. The Fung-Guccione model was used to describe myocardium mechanical behavior neglecting the active component. Myocardi fiber orientation was defined following the rule-based algorithm proposed by [PERSON] et al., based on the fibrous structure of canine hearts [261]. Boundary conditions consisted of uniform pressure at the endocardi and constraints to the displacement of the basal plane. Image-based endocardial and pericardial surfaces were projected into finite element model predictions for each time step and the mean-squared error was minimized by fitting the passive material properties. Results showed no significant differences in ventricular stiffness between groups, although patients with reduced ejection fraction presented elevated diastolic stress levels. [PERSON] et al.