Using the convexity of each component of the forward operator, we propose an extended primal-dual algorithm framework for solving a kind of nonconvex and probably nonsmooth optimization problems in spectral CT image reconstruction. Following the proposed algorithm framework, we present six different iterative schemes or algorithms, and then establish the relationship to some existing algorithms. Under appropriate conditions, we prove the convergence of these schemes for the general case. Moreover, when the proposed schemes are applied to solving a specific problem in spectral CT image reconstruction, namely, total variation regularized nonlinear least-squares problem with nonnegative constraint, we also prove the particular convergence for these schemes by using some special properties. The numerical experiments with densely and sparsely data demonstrate the convergence and accuracy of the proposed algorithm framework in terms of visual inspection of images of realistic anatomic complexity and quantitative analysis with metrics structural similarity, peak signal-to-noise ratio, mean square error and maximum pixel difference. We analyze the computational complexity of these schemes, and discuss the extended applications of this algorithm framework in other nonlinear imaging problems.