## 6 Applications to Cardiovascular Medicine
### The Heart
#### 6.2.1 Properties of the Healthy and Infarcted Ventricular Wall
According to this, myocardium behaves as an incompressible tissue only during rapid and reduced ejection and isovolumetric relaxation stages, while showing some degree of compressibility in the atrial systole, isovolumetric contraction, and filling stages. These observations are in agreement with in vivo compressibility measurements in large mammals [134]. A more sophisticated inverse analysis was presented by [PERSON] et al., in two consecutive papers published in 2011, introducing patient-specific geometries [255, 256]. MRI-based models of the LV at end-diastole were assumed as the zero-strain reference and discretized with Hermit-cubic finite elements. MRI tissue tagging was used to assess the diastole-to-systole displacement distribution and then interpolated into the nodes of the FEM mesh. The Fung-Guccione constitutive equation was selected to model the uniform passive properties of the myocardium. Myofiber orientations were assigned with a rule-based algorithm from [PERSON] et al. [55] based on canine serial histology from [PERSON] et al. [259]. Boundary conditions consisted of catheter measured ventricular pressure increments, zero traction at the epicardium, and apical and base displacement from tissue tagging. The least squared nodal displacement error was minimized using a reduced-order Kalman filter. The method was applied to one healthy heart and two patients with diastolic heart failure with impaired ejection fraction. The authors found a large difference in material parameters between healthy and heart failure patients, although the authors recognized the results were likely not unique for the given dataset. In addition, they found that passive behavior alone could not fully describe the deformation state at early diastole. They addressed this issue by introducing a time-dependent homogeneous active tension model and the backward displacement method to estimate the unloaded configuration [47]. Two different minimization problems are solved iteratively: first, the estimation of passive properties by consecutive simulation of deflation from early diastole to unloaded configuration followed by inflation to end-diastole; second the estimation of active properties by inflation from late diastole to systole (Figure 12b). The target minimization function for both iterative loops was defined as the error in nodal coordinates between simulations and interpolated tissue-tagging measurements for their corresponding end of process configuration. According to their results, the residual activation state from early to end-diastole was larger for patients with heart failure, which may indicate that diastolic relaxation is impaired after cardiac failure due to the compensation mechanism to maintain cardiac function. [PERSON] et al. applied methods to define patient-specific anatomic models and define active and passive behavior similar to [PERSON] et al. [47], although with no available patient-specific pressure data and the assumption of end-diastole as the stress-free reference [110]. Boundary conditions constrained all displacement on the basal plane of the ventricle and the dynamic boundary condition was a prescribed volume change, instead of the pressure increment. In the absence of pressure data, [PERSON] et al. assumed a well-established normalized LV pressure-volume curve of [PERSON] et al. [109] as the optimization target. MRI tissue tagging measurements were used to validate the converged results, which showed good agreement with image-derived circumferential and axial strain. Regarding fiber stress distribution, the authors found that the end of diastole myofiber stress peaked near the subendocardial wall. They also highlighted that the transmural variation of the end-of-systole myofiber stress was nonmonotonic and was maximal at the mid-wall of the ventricle. Solution multiplicity has been one of the main concerns about inverse methods, which motivated [PERSON] et al.