Due to developments in additive manufacturing (AM) techniques, design and producing cellular structures with complex topologies accompanied with appropriate mechanical properties and lightweight have become possible and the application of cellular porous materials has been increasing in various areas. In the current study, a novel cellular structure with adjustable radially graded relative density and properties inspired by bone tissue structure is designed and introduced. The cellular structure has five layers and is achieved by repeating a regular four-sided unit-cell in radial, peripheral, and axial directions by a specific pattern. Next, using analytical relations, the mechanical properties of the structure are derived. The obtained theoretical solution is validated by numerical modeling and experimental test of a polymeric specimen manufactured by SLA method. Comparison of the results shows good precision of the theoretical solution. Furthermore, the effect of design parameters including the height of representative volume element, the number of the sides of start shape, and radius of the struts on mechanical properties and their distribution is studied. Using genetic algorithms single-objective and multi-objective optimization is performed on elastic properties of the structure. The single-objective optimization results for structure with 70, 75, and 80% porosities led to 32.9, 35.92, and 35.68% improvement of elastic modulus to mass, respectively and 116.35, 96.48, and 73.62% increase of yield strength to mass at similar porosities compared to base models with same porosities. The results show proper ability of the structure in creating distribution of mechanical properties and porosity and its potential capability for use in bone replacement applications.