We propose a framework for the interactive simulation of nonlinear deformable objects. The primary feature of our system is the seamless integration of deformable simulation and collision culling, which are often independently handled in existing animation systems. The bridge connecting them is the medial axis transform or MAT, a high-fidelity volumetric approximation of complex 3D shapes. From the physics simulation perspective, MAT leads to an expressive and compact reduced nonlinear model. We employ a semi-reduced projective dynamics formulation, which well captures high-frequency local deformations of high-resolution models while retaining a low computation cost. Our key observation is that the most compelling (nonlinear) deformable effects are enabled by the local constraints projection, which should not be aggressively reduced, and only apply model reduction at the global stage. From the collision detection/culling perspective, MAT is geometrically versatile using linear-interpolated spheres (i.e. the so-called medial primitives) to approximate the boundary of the input model. The intersection test between two medial primitives is formulated as a quadratically constrained quadratic program problem. We give an algorithm to solve this problem exactly, which returns the deepest penetration between a pair of intersecting medial primitives. When coupled with spatial hashing, collision (including self-collision) can be efficiently identified on the GPU within few milliseconds even for massive simulations. We have tested our system on a variety of geometrically complex and high-resolution deformable objects, and our system produces convincing animations with all the collisions/self-collisions well handled at an interactive rate.