A parametrized mesoscale model for the early stage growth of isolated single or multiwall carbon nanotubes (CNTs) has been developed in order to investigate the effects of metal catalyst particle size and composition on CNT growth mechanism during synthesis via a substrate-supported, catalytic chemical vapor deposition process. The model is based on a coarse-grained graphene sheet, represented by a two-dimensional simply connected triangular mesh, with parameters for the surface curvature, bond stretching, carbon-carbon interaction, and carbon-catalyst interaction determined by classical molecular dynamics simulations using a bond-order potential derived from ab initio calculations. The mesoscale simulations show that the initial type of CNT growth is strongly influenced by the surface interaction energy between the graphene sheet and metal catalyst particle, rate of carbon deposition, and particle size. As expected, single wall tubes are produced from small catalyst particles at low deposition rates, but increasing the strength of carbon-catalyst interaction energy or carbon deposition rate results in double or even multiwall CNT structures, formed by folding or involution of the graphene sheet. For the range of model parameters investigated, all single wall CNTs with a diameter greater than 6.6 nm exhibited a kink-collapse transition once a certain critical tube length was reached.