Recently, several topology optimization methods in the Lagrangian or Eulerian description has accepted a wide of discussions, which have been applied to several design problems. This paper proposes a Isogeometric topology optimization with the Material Point Method (ITO-MPM) within a Lagrangian-Eulerian context, aiming to combine the unique strengths of both approaches for more effective optimization. First, an Isogeometric Material Points Method (I-MPM) is utilized to develop a numerical framework that solves static equilibrium equations while maintaining consistency between the geometric and analysis models through the use of shared NURBS (Non-Uniform Rational B-Splines) basis functions. Next, a Lagrangian-Eulerian particle moving model is introduced to represent structural topology in the optimization process. This model includes essential components such as the mappings of physical data (P2C: Particles to Control points, C2G: Control points to Gauss quadrature points) and the inverse mappings for sensitivity information (G2C and C2P). The ITO-MPM formulation for maximizing structural loading capability is then developed, where the positions and material consumption of particles are treated as design variables to guide the evolution of the topology. A detailed sensitivity analysis of the objective and constraint functions, based on particle-level physical data, is also presented. Finally, numerical design examples are provided to validate and demonstrate the effectiveness and advantages of the ITO-MPM approach, highlighting the critical role of particle movement in optimization.