Recent advances in medical imaging, computational methods, and biomechanics hold great promise for engineering-based decision making in clinical practice. Towards patient-specific modeling, however, we need to synthesize better the separate advances in computational biofluid mechanics and arterial wall mechanics. In this paper, we propose a mathematical model of growing fusiform aneurysms that is able to test multiple competing hypotheses with regard to the production, removal, and organization of intramural collagen, and thus to predict their consequences in enlargement and changes in material properties of the lesion. To apply this model to realistic cases, including fluid-solid interactions, we also need to develop a method to exploit current advances in computational biofluid mechanics. Thus, we describe a method to represent highly nonlinear and anisotropic material behaviors within a linearized constitutive equation commonly employed in fluid-structure simulations of blood flow in deformable arteries.
Modeling aneurysm growth using stress-mediated growth laws requires accurate knowledge of the hemodynamic stresses and strains acting on the aneurysm wall due to the internal blood flow and the external tissue support. Therefore, solving the coupled problem of blood flow and vessel wall deformation represents a critical step in the evaluation of these hemodynamic stresses, but for large, patient-specific models of the vasculature one that is computationally expensive. In this work, we present the application of a new formulation, the Coupled Momentum Method for Fluid-Solid Interaction (CMM-FSI), to compute blood flow and vessel wall deformation under realistic ranges of pressures for large patient-specific models of the cerebro-vasculature. The method couples the equations of the deformation of the vessel wall at the variational level as a boundary condition for the fluid domain. We consider a strong coupling of the degrees-of-freedom of the fluid interface and the wall domains. The effect of the vessel wall boundary is therefore added in a monolithic way to the fluid equations, resulting in a remarkably robust and computationally-efficient scheme. The method is applied to patient-specific model of the Circle of Willis featuring a saccular aneurysm, using resistance outflow boundary conditions. The wall normal and shear stresses resulting from the simulation can then be used as the hemodynamic forces mediating the aneurysm wall adaptation in the algorithm shown in the second part of this work.