A better understanding of the density dependence of the nuclear equation of state (EOS) of isospin asymmetric nuclear matter, particularly at high den- sities, is crucial for interpreting many astrophysical phenomena e.g. neutron stars and supernovae, where model descriptions require inputs for the behaviour of the EOS with extreme values of the symmetry energy. The nuclear EOS is a fundamental property of nuclear matter, which describes the relationships between its thermodynamic properties but is insufficiently constrained by ex- periment. Indeed the equation of state of asymmetric nuclear matter is poorly understood, largely due to a lack of knowledge regarding the density depen- dence of the nuclear symmetry energy term, especially at densities considerably greater than that of normal nuclear matter. Theoretical studies based on either microscopic many-body theories or phenomenological approaches give widely divergent predictions of this quantity. While considerable progress has been achieved over recent years in determining the symmetry energy at saturation and sub-saturation nuclear matter densities, much more work is still needed to probe its behaviour at supra-saturation densities. Until recently, there has been little possibility to study experimentally the density dependence of the symmetry energy. This situation is changing dramatically as new radioactive beam facilities become operational. These facilities promise to revolutionise our knowledge of the equation of state of asymmetric nuclear matter. Heavy- ion reactions induced by neutron-rich nuclei have thus a crucial role to play in this field of research. They provide a unique tool for obtaining information concerning the high-density behaviour of the symmetry energy in a laboratory environment. This thesis reports on an experimental feasibility study into the use of observables of collective flow measured in relativistic heavy-ion collisions to investigate the symmetry energy at supra-saturation densities.