This paper addresses the stability problem of a biological system that describes the proliferation of sick cells in Acute Myeloid Leukemia (AML). AML therapies aim at eradicating malignant cells, reaching a biological status represented by the zero equilibrium point of the age-structured mathematical model describing pathological hematopoeisis. First, the AML stability problem is reformulated into a stability problem of a nonlinear cascaded system. Then based on a positivity property of the system, non quadratic Lyapunov candidates are constructed. Finally, necessary and stability conditions are obtained. These conditions complete and generalize previous results where the main contribution consists in providing necessary and sufficient conditions based on a general model that incorporates fast self renewal. This model is complex but more realistic from a practical pint of view. Further more, unlike previously published works, the proposed conditions do not depend on auxiliary parameters which are biologically ambiguous but depend only on the AML system which makes the results more biologically relevant for AML treatment.