A 16-moment bi-Maxwellian model to describe the proton temperature anisotropy and heat flux density in the fast solar wind is presented. Although the proton total heat flux density in this model is found to be significantly smaller than the classical Coulomb collision-dominated Spitzer-H¿rm heat flux density, it still plays a significant role in shaping the proton temperature anisotropy. The proton total heat flux density matches the value observed between 0.3 and 1 AU when the proton temperature is also close to the observed value. A large temperature anisotropy in the inner corona, with a proton perpendicular temperature higher than the parallel temperature, as indicated by recent ultraviolet observations, leads to a higher perpendicular heat flux. Since the perpendicular heat flux conducts energy from the perpendicular to the parallel degree of freedom, an unrealistically high parallel temperature develops in interplanetary space if no cooling mechanism for protons in the direction parallel to the magnetic field is introduced. This study suggests that with a bi-Maxwellian-based 16-moment description of the solar wind protons the high proton perpendicular temperature implied by the Ultraviolet Coronagraph Spectrometer on board Solar and Heliospheric Observatory in the inner corona and by in situ measurements from Helios place strong constraints on the proton heating mechanism: not only must the protons be heated in the direction perpendicular to the magnetic field, but they must also be cooled in the parallel direction. ¿ 1999 American Geophysical Union