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This paper presents a detailed study of the oscillation characteristics of a bubble confined inside a deformable microvessel, whose size is comparable with the bubble size. The vessel's compliance is characterized by a nonlinear relation between the intraluminal pressure and the expansion ratio of the vessel radius, which represents the variation of the vessel stiffness with the pressure of the filling liquid. In this analysis, an initially spherical bubble evolves into an ellipsoid, and the asymmetric oscillation appears immediately after the driving pressure is applied and magnifies with oscillation cycles. Compared with the symmetric oscillation in an unconstrained environment, the vessel constraint makes the bubble contract significantly more and subsequently expand in a more violent rebound, inducing substantially larger peaks of the intraluminal pressure exerted on the vessel wall. A larger initial bubble/vessel radius ratio leads to not only a larger peak but also a higher oscillation frequency of the intraluminal pressure, which are the two most dominating parameters in determining the vessel's failure under cyclic loading. The numerical results have further shown that an increase of the vessel wall stiffness strengthens the asymmetric effect, i.e., a larger peak of the intraluminal pressure with a higher oscillation frequency, and so does a larger pre-existing pressure in the liquid filling the vessel. These findings imply that the asymmetric effect is one of the primary mechanisms for clinical injuries of capillary and small blood vessels and for the higher risk of pediatric and hypertension patients in shock wave lithotripsy.