Interior tomography reconstructs a region of interest using truncated projection data, but it is subject to the ill-posedness of interior tomography. With the photon-counting detector, K-edge imaging uses data in the low and high energy bins around the K-edge of a contrast agent, and can faithfully recover true image contrast for improved diagnosis. The purpose of this paper is to reconstruct a region of interest inside a patient assuming the existence of a known K-edge material. In this case, there is a significant difference in x-ray attenuation around the K-edge, but these attenuation coefficients are inter-related to guide updating an intermediate reconstruction until a stopping criterion is satisfied. In our study, new interior tomography algorithms were developed without any major computational overhead, and several phantoms were used to validate the algorithms. The proposed methods are advantageous relative to the existing interior tomography algorithms, because of the available spectral information in the form of a known K-edge material.