## 6 Applications to Cardiovascular Medicine
### Heterogeneous Models  
Modeling of material heterogeneity can provide better fits to kinematic data, resolve property changes, and identify the location and severity of myocardial lesions. However, this comes with an increased modeling effort and computational expense. By assuming material heterogeneity on inverse methods, the number of parameters to be fitted increases, posing a burden on the optimization algorithm and complicating the solution of the forward problem. A common approach is to approximate spatial variations of myocardium properties and microstructure with region-wise heterogeneities. The simulation domain of the myocardium is divided into segments, each one with its own set of homogeneous material properties. The American Heart Association (AHA) proposed the division of the left ventricle into 17 standardized LV segments which have been adopted extensively in the study of myocardium mechanics (Figure 12a) [262]. One of the earliest inverse analyses of biventricular models with region-wise material heterogeneity was proposed by [PERSON] et al. [254]. The study gathered cine MRI datasets from 8 healthy volunteers and 3 heart failure patients with impaired ejection fractions. Cine MRI was used to estimate the volume change of both ventricles, to identify the location of the epicardial surface on several time-steps over the cardiac cycle, and to estimate displacements with a diffeomorphic free form deformation algorithm. End-diastole was used as the reference configuration and to build a deformable FEM mesh. Electromechanical behavior was modeled with a Bestel-Clement-Sorine model, which consists of a Mooney-Rivlin hyperelastic material matrix reinforced with fibers with passive elastic and time-dependent active components. Fiber orientation was assumed to follow the Laplace-Dirichlet rule-based algorithm by [PERSON] et al. [55]. The active component was assumed to have a viscous dissipation component and was modeled by a two-differential equation system solving for the time-dependent active stress and sarcomere stiffness as a function of an activation state variable. Parameter fitting is carried out by applying Kalman filters in two steps: first, a general fit is achieved with the overall pressure-volume curve, followed by a parameter refinement for each sector using sector-specific displacements and change of LV section volume. The model was able to locate the infarcted regions by assigning them lower contractility, while healthy patients converged to more homogeneous property distributions and normal active function. In 2017, [PERSON] et al. performed an inverse analysis on 27 healthy subjects and 11 patients with acute myocardial infarction [253]. Gadolinium-enhanced and cine MRI were applied to identify the location of infarcted regions and to calculate the volume change of the LV. Anatomic models were built at end-diastole, which was assumed as the zero-strain reference. The LV systolic blood pressure was approximated by the sphygmomanometer systolic measurements. The anatomic models were divided into the 17 standard AHA regions, and circumferential strains were calculated for each region through a b-spline deformable registration algorithm. Non-infracted tissue was modeled as a Holzapfel-Ogden material with a sophisticated differential-algebraic model for active stress. Myofiber orientation was defined by the minimum-distance rule-based algorithm [PERSON] et al. [56]. Infracted tissue was assumed 50-fold stiffer than regular tissue with no active contraction. A Bayesian approach with Gaussian processes and automatic relevance determination algorithm was used to fit material properties and active contraction parameters by minimizing a weighted function of the volume error and region-wise circumferential strain error. Results showed that active tension was larger in infarcted hearts, which agrees with the early observations of [PERSON] et al. and [PERSON] et al., for which the authors hypothesized the existence of a compensation mechanism for infarcted hearts to preserve stroke volume.