Blood vessels are commonly studied in isolation to define their mechanical and biological properties under controlled conditions. While sections of the wall are sometimes tested, vessels are most often attached to needles and examined in their natural cylindrical configuration where combinations of internal pressure and axial force can be applied to mimic in vivo conditions. Attachments to needles, however, constrain natural vessel response, resulting in a complex state of deformation that is not easily determined. As a result, measurements are usually limited to the midsection of a specimen where end effects do not extend and the deformation is homogeneous. To our knowledge, however, the boundaries of this uninfluenced midsection region have not been explored. The objective of this study was to define the extent of these end effects as a function of vessel geometry and material properties, loading conditions, and needle diameter. A computational fiber framework was used to model the response of a nonlinear anisotropic cylindrical tube, constrained radially at its ends, under conditions of axial extension and internal pressure. Individual fiber constitutive response was defined using a Fung-type strain energy function. While quantitative results depend on specific parameter values, simulations demonstrate that axial stretch is always highest near the constraint and reduces to a minimum in the uninfluenced midsection region. Circumferential stretch displays the opposite behavior. As a general rule, the length of the region disturbed by a needle constraint increases with the difference between the diameter of the needle and the equilibrium diameter of the blood vessel for the imposed loading conditions. The reported findings increase the understanding of specimen deformation in isolated vessel experiments, specifically defining considerations important to identifying a midsection region appropriate for measurement.