## 6 Applications to Cardiovascular Medicine
### The Heart
#### 6.2.1 Properties of the Healthy and Infarcted Ventricular Wall  
The ventricular wall is a complex multilayered composite responsible for delivering the driving force to pump blood throughout the cardiovascular system. The myocardium is the functional layer of the ventricular wall, containing the myofib responsible for the active contraction of the muscle and the structural collagen fibers that contribute to its bulk mechanical properties. An accurate understanding of myocardial mechanics is key for the diagnosis and treatment of diverse cardiac pathologies, and potentially, predicts and stratifies the risk of heart failure after infarct. Therefore, many studies have focused on the estimation of mechanical properties of healthy myocardium, and more interestingly, estimating the effects of ischemia, and quantifying the properties of infarcted cardiac tissue to yield a truly patient-specific risk assessment of cardiac failure. Most developments relied on FEM for the solution of a forward problem (summarized in Table 7). \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Study** & **Clinical Data** & & & **Forward Problem** & & **Inverse Problem** \\ \hline \multirow{4}{*}{[PERSON]} & Population Pathology & 12 H & Reference & End of diastole & Target & Least squared error to \\  & Pathology & None & Passive model & Hom. Guccione & function & Klotz P-V \\  & Data & Cine MRI & Active model & 1 eq. active stress & Opt. & Nelder Mead. \\  & Anatomy & LV with RBFO by & Boundary & ICP, TF epicardium & algorithm & Nelder Mead. \\ \hline \multirow{4}{*}{[PERSON]} & Population Pathology & 1H 5D & Reference & Early diastole & Target & Volume change error and \\  & Pathology & FMR-CAD & Passive model & Het. Guccione & function & segment-wise strain error \\  & Data & Cine MRI, IT, Stress perfusion MRI, GE MRI, 4D US, Hand cuff pressure & Active model & 2 eq. active stress & & \\  & Anatomy & BV in 17 AHA regions with RBFO by [PERSON] et al. & Boundary & ICP, TF epicardium, Constrained base & Opt. & \\ \hline \multirow{4}{*}{[PERSON] et al., 2018 [141]} & Population Pathology & 1D & Reference & End of diastole & Target & Deformation gradient \\  & Pathology & LBBB and CI & Passive model & Het. Holzapfel-Ogden & function & error. \\  & Data & 4D US, USST, GE MRI, ICP & Active model & None & & Sequential quadratic \\  & Anatomy & LV in 17 AHA regions with RBFO by [PERSON] et al. & Boundary & ICP, Constrained apex, EF at base. & Opt. & \\ \hline \multirow{4}{*}{[PERSON] et al., 2018 [248]} & Population Pathology & 5H 19D & Reference & Diastasis & Target & Least-squared error to P-V \\  & Pathology & HFFE, HPEPEP & Passive model & Hom. Guccione & function & curve. \\  & Data & Cine MRI, ICP CV with RBFO by & Active model & None & & Opt. \\  & Anatomy & [PERSON] et al. & Boundary &  
The assumption of material homogeneity is a common and convenient simplification for forward and inverse models. It limits the number of parameters to be fit while still reproducing the overall mechanics of the organ with reasonable accuracy. Even though the myocardium is highly complex and spatially heterogeneous, homogeneous models may be deemed to be adequate for the study of healthy hearts, or when the aim of the analysis is not centered on the study of focalized lesions. In the study of tissues with steep localized changes in structure and properties, as in infarcted myocardium, the material homogeneous models cannot reproduce the localized strain and stiffness distributions on the infarct itself and the infarct borderzone, providing only averaged estimations of local deformation and material properties. However, these averaged properties can still be used as a measure of lesion severity by comparative studies of healthy cases. One of the first simplified inverse models of the left ventricle (LV) was introduced by [PERSON] et al., to study the compressibility of the myocardium [257].