In diagnostic ultrasound imaging, the image reconstruction quality is crucial for reliable diagnosis. Applying reconstruction algorithms based on the acoustic wave equation, the obtained image quality depends significantly on the physical material parameters accounted for in the equation. In this contribution, we extend a proposed iterative nonlinear one-parameter compressibility reconstruction algorithm by the additional reconstruction of the object's inhomogeneous mass density distribution. The improved iterative algorithm is able to reconstruct inhomogeneous maps of the object's compressibility and mass density simultaneously using only one conventional linear transducer array at a fixed location for wave transmission and detection. The derived approach is based on an acoustic wave equation including spatial compressibility and mass density variations, and utilizes the Kaczmarz method for iterative material parameter reconstruction. We validate our algorithm numerically for an unidirectional pulse-echo breast imaging application, and thus generate simulated measurements acquired from a numerical breast phantom with realistic compressibility and mass density values. Applying these measurements, we demonstrate with two reconstruction experiments the necessity to calculate the mass density in case of tissues with significant mass density inhomogeneities. When reconstructing spatial mass density variations, artefacts in the breast's compressibility image are reduced resulting in improved spatial resolution. Furthermore, the compressibility relative error magnitude within a diagnostically significant region of interest (ROI) decreases from 3.04% to 2.62%. Moreover, a second image showing the breast's inhomogeneous mass density distribution is given to provide additional diagnostic information. In the compressibility image, a spatial resolution moderately higher than the classical half-wavelength limit is observed.