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The three-dimensional characterization of magnetic flux-ropes observed in the heliosphere has been a challenging task for decades. This is mainly due to the limitation to infer the 3D global topology and the physical properties from the 1D time series from any spacecraft. To advance our understanding of magnetic flux-ropes whose configuration departs from the typical stiff geometries, here we present the analytical solution for a 3D flux-rope model with an arbitrary cross-section and a toroidal global shape. This constitutes the next level of complexity following the elliptic-cylindrical (EC) geometry. The mathematical framework was established by Nieves-Chinchilla et al. (2018) ApJ, with the EC flux-rope model that describes the magnetic topology with elliptical cross-section as a first approach to changes in the cross-section. In the distorted-toroidal flux rope model, the cross-section is described by a general function. The model is completely described by a non-orthogonal geometry and the Maxwell equations can be consistently solved to obtain the magnetic field and relevant physical quantities. As a proof of concept, this model is generalized in terms of the radial dependence of current density components. The last part of this paper is dedicated to a specific function, $F(φ)=δ(1-λ\cosφ)$, to illustrate possibilities of the model. This model paves the way to investigate complex distortions of the magnetic structures in the solar wind. Future investigations will in-depth explore these distortions by analyzing specific events, the implications in the physical quantities, such as magnetic fluxes, heliciy or energy, and evaluating the force balance with the ambient solar wind that allows such distortions.