  The design of an experiment, e.g., the setting of initial conditions,
strongly influences the accuracy of the whole process of determining model
parameters from data. We impose a sensitivity-based approach for choosing
optimal design variables and study the optimization of the shape (and topology)
of the initial conditions for an inverse problem of a diffusion parameter
identification. Our approach, although case independent, is illustrated at the
FRAP (Fluorescence Recovery After Photobleaching) experimental technique. The
core idea resides in the maximization of a sensitivity measure, which depends
on a specific experimental setting of initial conditions. By a numerical
optimization, we find an interesting pattern of increasingly complicated (with
respect to connectivity) optimal initial shapes. The proposed modification of
the FRAP experimental protocol is rather surprising but entirely realistic and
the resulting enhancement of the parameter estimate accuracy is significant.
