Partial volume effects are often experienced in diffusion-weighted MRI of biologic tissue. This is when the signal attenuation reflects a mixture of diffusion processes, originating from different tissue compartments, residing in the same voxel. Decomposing the mixture requires elaborated models that account for multiple compartments, yet the fitting problem for those models is usually ill posed. We suggest a novel approach for stabilizing the fitting problem of the multiple-tensors model by a variational framework that adds biologically oriented assumption of neighborhood alignments. The framework is designed to address fiber ambiguity caused by a number of neuronal fiber compartments residing in the same voxel. The method requires diffusion data acquired by common, clinically feasible MRI sequences, and is able to derive familiar tensor quantities for each compartment. Neighborhood alignment is performed by adding piece-wise smooth regularization constraints to an energy function. Minimization with the gradient descent method produces a set of diffusion-reaction partial differential equations that describe a tensor-preserving flow towards a best approximation of the data while maintaining the constraints. We analyze fiber compartment separation capabilities on a synthetic model of crossing fibers and on brain areas known to have crossing fibers. We compare the results with diffusion tensor imaging analysis and discuss applications for the framework.