With the progressive inclusion of numerical simulations in medical research and clinical practice, accuracy and reliability of patient-specific computational analyses need to be properly certified. This raises new challenges when estimating patient-specific parameters that may be too difficult or even impossible to measure in practice. On the other hand, these parameters represent a macroscale synthesis of molecular or mesoscale dynamics, but their practical individual-based quantification based on modeling arguments is extremely difficult. Data assimilation techniques are required to merge available data and numerical models to assess the reliability of a quantitative analysis. In this talk, variational procedures will be considered to estimate (a) vascular compliance from available measures of displacement; (b) cardiac conductivities from available measures of cardiac potentials. Some theoretical as well as practical aspects of the numerical solution of these problems will be addressed. In particular, we pursue variational techniques based on a constrained minimization approach, the constraint being represented by the fluid-structure interaction vascular problem or by the Bidomain equations for electrocardiology. We will discuss several technical details of this approach. In general, these techniques lead to high computational costs and proper methods for the sake of computational efficiency need to be adopted. We consider in particular both methods based on simplified models for the forward problem (like the Monodomain equation) or on surrogate solutions obtained on the basis of the offline/online paradigm, like the Proper Orthogonal Decomposition method (POD). We will illustrate both succesfull experiences as well as pitfalls of these approaches.