The two-part model is often used to analyse medical cost data which contain a large proportion of zero cost and are highly skewed with some large costs. The total medical costs over a period of time are often censored due to incomplete follow-up, making the analysis difficult as the censoring can be informative. We propose to apply the inverse probability weighting method on a two-part model to analyse right-censored cumulative medical costs with informative censoring. We also introduce a set of simple functionals based on the intermediate cost history to be applied with the efficiency augmentation technique. In addition, we propose a practical model-checking technique based on the cumulative residuals. Simulation studies are conducted to evaluate the finite sample performance of the proposed method. We use a data set on the cardiovascular disease (CVD)-related Medicare costs to illustrate our proposed method.