## 6 Applications to Cardiovascular Medicine
### The Unloaded Reference Configuration in Cardiovascular Mechanics
One of the first contributions was proposed by [PERSON] et al., who developed an optimization framework for an arbitrary parameter k such that the coordinates of the unknown zero-pressure reference geometry (\(x_{0}\)) can be approximated by the difference of the in vivo deformed configuration (\(x_{i}\)) minus k times the displacement produced by the pressure load on that configuration (\(\mathbf{U}\)), i.e., \(x_{0}=x_{i}-k\mathbf{U}\). The main conceptual limitation of this method is the assumption that the backward deformation field is linearly related to the forward deformation field through the factor k. This method was then applied to estimate the unloaded configuration of a patient-specific AAA [236]. The backward displacement method was introduced by [PERSON] et al., in 2007, for breast biomechanics and by [PERSON] et al., in 2013, for cardiovascular tissue [234, 237]. This method solves the unloaded configuration using the fixed-point interactions proposed by [PERSON] et al. [100]. It consists in approaching the zero-pressure geometry by iteratively updating the reference configuration, calculated by subtracting the nodal displacement vector between the updated deformed configuration and the target in vivo configuration until a required error tolerance is reached. [PERSON] et al. successfully applied a similar pullback algorithm to 12 patient-patient specific models of AAA built from CT scans which were assumed to be at uniform diastolic pressure on the image-based deformed geometry. They tested isotropic and anisotropic material models, assuming material homogeneity with reference material constants from the literature [238]. [PERSON] et al. proposed an augmented Sellier's method based on [PERSON]'s delta-squared process, by introducing an augmentation parameter to accelerate the convergence rate and increase the chances of convergence. The method was applied to find the unloaded geometries of a thrombus and heart valve leaflets from animal models with geometries and properties collected from previous studies [239]. More recently, [PERSON] et al. proposed the shrink-and-fit algorithm, that assumes the unloaded configuration is a shrink analogous to the loaded reference geometry. On each iteration step, the coordinates of each node are mapped into a smaller geometry affected by a circumferential and axial shrink factor, the new geometry is loaded by the reference pressure until the least squared error of the nodal coordinates of the reference and inflated model is minimized. The method was then applied to resolve the unloaded configuration of an ideal and a patient-specific artery model assuming Mooney-Rivlin hyperelastic behavior and employing the Nelder-Mead optimization algorithm [240]. A different iterative approach is to solve the strain and stress distribution that balances the applied loads acting on the image-derived anatomic configurations without the resolution of the unloaded geometry [232]. Some methods that fit this category are the backward incremental algorithm and the modified updated Lagrangian formulation. In the backward incremental algorithm, small increments of the pressure load are applied to the reference geometry, the resulting stress state is mapped to the reference geometry as the initial condition for the next pressure increment until static equilibrium is reached. This method was applied by [PERSON] et al., using patient-specific AAA geometries and pressure loads to determine the stress distribution at diastole while assuming isotropic Neo-Hookean material behavior with population-averaged constants [241]. Similarly, the modified updated Lagrangian formulation applies consecutive small loads increments on the image-based reference configuration to build up an incremental multiplicative update  
Figure 11: Results and accuracy of the direct inverse elastostatic problem implemented on a FEM solver and applied to a porcine biventricular model. (**a**) Reference loaded configuration reconstructed from MRI scans.