Some of the challenges in cardiovascular mathematics. The increasing power of mathematical models and methods, together with the in- creasing computational power of modern architectures, has led cardiovascular math- ematics to become not only a descriptive tool, but more importantly, a tool for diag- nosis and treatment design [74, 93]. In silico experiments (that is, numerical simu- lations of possible scenarios) have become an extremely useful resource to analyze a pathology and investigate possible solutions. The advantage of in silico experi- ments compared to in vitro experiments is twofold: i) it does not require invasive testing, which for certain patologies and certain regions of the human body may be dangerous, and ii) when investigating possible treatments, it allows to consider and compare multiple solutions before they are concretely implemented. This not only reduces the risk of experimental treatments, but sometimes also their costs (for instance, when developing new devices). However, in order for the numerical results to really help the diagnosis and treat- ment of the pathology of a patient, two requirements must be satisfied: • the inputs of the mathematical model (such as geometry, parameters, bound- ary conditions) must be the ones corresponding to the current patient. • the numerical method must give a solution which is reliable. As we mentioned in the previous section, advancement in imaging techniques and image processing have made it possible to create geometries (i.e., computa- tional domains) that are specific to the patient at hand [72]. Nevertheless, some of the other inputs are usually not known, or known with limited precision. For in- stance, when simulating the motion of blood in a vessel, the full velocity (or stress) profile at the inflow is usually not known, and all that is available (from medical measurements) is some average quantity (such as the pressure or flow rate). An- other example is given by the mechanical properties of the vessel: in a linearly elas- tic framework, these are condensed into two parameters, the Young’s modulus and the Poisson’s ratio. When simualting a blood flow in a compliant vessel, in order to solve the FSI problem arising from the interaction between blood and vessel wall, these parameters must be known. If the patient is known to be in “good” condition, usually an average value taken from literature is used. This not only gives up the specificity of the numerical experiment, but it may give misleading results in case the patient is in fact ...
