The beam width of Laguerre-Gaussian vortex beam is derived by using intensity momentum method. Using the expression of Hermite-Gaussian beam in terms of spherical vector wave functions by complex source point method, and combining with the transformation between Laguerre polynomial and Hermite polynomial, Laguerre-Gaussian vortex beam is expanded in terms of spherical vector wave functions. Utilizing the generalized Lorenz Mie theory, the scattering of Laguerre-Gaussian vortex beam by cancer cells is investigated. The intensity and phase distribution at cross section perpendicular to the propagation axis with different beam modes and different propagation distances is analyzed numerically. The results show that with the increase of the radial and azimuthal mode index and the propagation distance, the size of the beam increases. The phase distribution of Laguerre-Gaussian vortex presents different characteristics from source plane with the increase of propagation. Stemming from the intensity distribution of Laguerre-Gaussian vortex beam, the scattering properties are interpreted.