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Chambolle-Pock (CP) algorithm is a first-order primal-dual algorithm for convex optimization problems. It has been used in CT and PET image reconstruction based on linear data model. Non-convex optimization programs, such as those based on a non-linear data model in multispectral CT, can not be solved by applying the CP algorithm. In this work, we propose a non-convex CP (ncCP) algorithm, inspired by our previous work on an algorithm for a non-convex optimization program of a specific form in multispectral CT. The proposed ncCP algorithm can deal with different optimization program designs including non-smooth objectives and/or constraints and also involves relatively fewer parameters for convergence. An algorithm instance for an example of non-convex optimization program used in multispectral CT reconstruction has been derived, together with the convergence conditions. A numerical study using computer simulation data with two spectra has been carried out for verifying the numerical convergence of the proposed ncCP algorithm.