Wall flow filters have successfully been used for many years to abate particulate matter emissions. Since then, modeling of this type of filter device has supported the development by a broad variety of approaches reaching from explicit pressure drop correlations up to complex 3D CFD simulations. 1D models are commonly used in the context of plant modeling supporting control development and calibration. Here most differences can be identified by the applied filtration, active or passive soot regeneration and catalytic reaction mechanisms. The proposed paper discusses a 1D+1D wall flow filter model resolving transport phenomena along the axial direction of the inlet/outlet channel and also in transverse direction through the soot cake and filter wall. The basic set of gas phase flow equations is extended by the passive transport of an arbitrary number of soot classes. The balance equations of deposit soot are extended to handle individual soot population. The filtration model considers the influence of operating conditions, the wall pore structure and the distribution of soot particle sizes. The soot oxidation model takes into account microstructural differences in the soot catalyst interaction. The filtration model is validated with the help of an analytical reference solution for empty filters. In addition, the impact of numerical discretization on model accuracy and computational effort is presented. Both models are also compared to experimental data revealing good agreement and a reasonable effort for model parameterization. Additionally different filtration approaches from literature are compared. The validated model is applied in two use-cases demonstrating its consistence from office to HiL simulation. In a first concept study, the interaction of soot filtration and soot oxidation is simulated for different operating conditions. Second, the filter model, as part of an entire exhaust line, is connected to a thermodynamic model of a turbocharged 2 liter high speed direct injection engine running in a passenger car vehicle model. On a HiL environment, a random drive cycle is selected to demonstrate the real-time capability of both the engine and aftertreatment plant model.