Modeling lithium-ion batteries is crucial for electrochemical energy storage to characterize their behavior and predict their State-Of-Charge and State-Of-Health. Equivalent Circuit Models can identify voltage and temperature profiles under a given input current with little computational cost. However, they cannot explain relevant microscopic phenomena that determine the battery's capability to deliver power and be efficiently charged. Conversely, Physics-Based Models can consider advanced physics and account for microscopic information at an increased cost, precluding their utilization in Battery Management Systems. This paper proposes to validate and integrate the two cited modelling approaches applied to a commercial lithium iron phosphate battery. The latter are validated with discharge-charge experimental tests, and microstructural data directly measured through the experimental disassembly of the battery. In this way, equivalent battery models become ideally linked to physics-based ones to fit all the internal electrochemical processes for a complete understanding of the evolution of battery states over time.