  We discuss the evaluation of the real part of the elementary amplitudes in
the context of a multiple diffraction model for $pp$ elastic scattering earlier
developed. The framework is based on the concepts of analyticity and polynomial
boundedness, and the techniques of dispersion relations. Novel results concern
the use of derivative dispersion relations at the elementary level
(constituent-constituent interactions) and an optimization of these relations
in terms of one free parameter. In addition to a theoretical improvement, we
achieved a satisfactory description of the physical quantities.
