The role of mathematics in understanding and simulating fluid dynamics and biochemical processes in the physiological and pathological functioning of the human cardiovascular system is becoming more and more crucial. These phenomena are indeed correlated with the origin of some major cardiovascular pathologies, and influence the efficacy of the treatments to heal the arteries from their diseases. Mathematical models allow the description of the complex fluid-structure interaction which govern the artery wall deformation under the pressure pulse. Moreover, appropriate reduction strategies can be devised to allow for an effective description of the interaction between large, 3D components, and small 1D branches of the circulatory system, as well as the transfer of drugs and chemicals between the arterial lumen and the vessel wall. This presentation will address some of these issues, the role of medical imaging and their integration in the numerical process as well as in the validation step.