## 6 Applications to Cardiovascular Medicine
### Hemodynamics
#### 6.5.1 Rigid Wall Models  
[PERSON] et al. applied an iterative inverse method for the hemodynamic study of three descending thoracic aortic aneurysms. CT scans were used to build the 3D models that included the ascending and descending aorta [118]. PC MRI sequences were collected at the ascending aorta (above the aortic bulb), the suprarenal abdominal aorta, and all three branches of the aortic arch. Diastolic and systolic pressures were collected from sphygmomanometry. The authors observed that balances with the inlet and outlet flow rates measured with PC MRI did not comply with the conservation of mass principle. The forward problem was defined by applying the PC MRI-derived velocity distribution in the ascending aorta as an inlet boundary condition. At all four outlets, a surrogate three element Windkessel model of unknown parameters was imposed as a boundary condition, while blood was assumed to be an incompressible Newtonian fluid. The forward problem was solved by a FEM solver. An optimization algorithm was applied to minimize the least-squared error of the measured blood flow at the outlets to simulation estimates, by fitting the surrogate model parameters. The authors highlight that this weak approach allows distributing the error related to measurement noise while enforcing mass conservation. Similarly, [PERSON] et al. used PC MRI data from a healthy volunteer to find the pressure gradient distribution in the circle of [PERSON], an important cerebral arterial system [299]. In this work, Kalman filters are iterated over pressure boundary conditions to adjust the simulated flows to PC MRI measurements with an FVM solver for the forward problem. Noise-to-signal ratios were used to incorporate the measurement uncertainty into the data analysis. The authors highlight that the outputs of the inverse methods yield smaller uncertainties than CFD or 4D flow MRI data analysis alone.  
[PERSON] et al. proposed a modification to the implementation of FVM for fluid dynamics problems, to introduce the minimization of simulated nodal velocity components to 4D flow data measurements in the linearized SIMPLER algorithm [300]. The method required the smoothing and interpolation of coarse 4D flow MRI data to the FVM mesh. 4D flow MRI-derived velocities were directly used as inlet and outlet boundary conditions. The minimization problem and FVM solution were solved simultaneously using a version of the iterative Runge-Kutta algorithm. This method allowed the simultaneous solution of the simulation and inverse problems, thus reducing the computational expense. As a proof of concept, the method was applied to anatomy and 4D flow MRI scans of a healthy human carotid artery. The method was incorporated into a custom-made solver that required special discretization into a structured mesh in the Cartesian space. [PERSON] et al. further developed this approach by incorporating the nodal velocity error minimization approach into a discontinuous Galerkin FEM formulation, allowing the solution of unstructured meshes [122]. The method was validated to in vitro measurements with laser particle imaging velocimetry in a pulsating flow loop with an abrupt change of cross-sectional area to induce complex flow patterns. Then, the method was applied to a healthy-human proximal cerebral artery. CT angiography was used to build the 3D anatomic model, 4D flow data were collected at a resolution of 0.7 mm voxel size with a 7 T scan, and PC MRI scans were collected at inlet and outlet planes with a resolution of 0.5 mm/px. Moreover, 4D flow data were spatially and temporally smoothed and interpolated into the FEM mesh, while PC MRI data were integrated to enforce inlet and outlet transient plug flow as boundary conditions. The method showed errors below 1% on velocity distribution for in vitro validation tests, and the proof of concept on in vivo datasets demonstrated the potential of the proposed methodology for future human studies.