A mathematical model given by a two-dimensional differential system is introduced in order to understand the transition process from the normal hematopoiesis to the chronic and accelerated-acute stages in chronic myeloid leukemia. A previous model of Dingli and Michor is refined by introducing a new parameter in order to differentiate the bone marrow microenvironment sensitivities of normal and mutant stem cells. In the light of the new parameter, the system now has three distinct equilibria corresponding to the normal hematopoietic state, to the chronic state, and to the accelerated-acute phase of the disease. A characterization of the three hematopoietic states is obtained based on the stability analysis. Numerical simulations are included to illustrate the theoretical results.

A mathematical model given by a two - dimensional differential system is introduced in order to understand the transition process from the normal hematopoiesis to the chronic and accelerated acute stages in chronic myeloid leukemia. A previous model of Dingli and Michor is refined by introducing a new parameter in order to differentiate the bone marrow microenvironment sensitivities of normal and mutant stem cells. In the light of the new parameter, the system now has three distinct equilibria corresponding to the normal hematopoietic state, to the chronic state, and to the accelerated acute phase of the disease. A characterization of the three hematopoietic states is obtained based on the stability analysis. Numerical simulations are included to illustrate the theoretical results.

In this paper, two control problems for a symmetric model of cell dynamics related to leukemia are considered. The first one, in connection with classical chemotherapy, is that the evolution of the disease under treatment should follow a prescribed trajectory assuming that the drug works by increasing the cell death rates of both malignant and normal cells. In the case of the second control problem, as for targeted therapies, the drug is assumed to work by decreasing the multiplication rate of leukemic cells only, and the control objective is that the disease state reaches a desired endpoint. The solvability of the two problems as well as their stability are proved by using a general method of analysis. Some numerical simulations are included to illustrate the theoretical results and prove their applicability. The results can possibly be used to design therapeutic scenarios such that an expected clinical evolution can be achieved.

Chronic myelogenous leukemia (CML) is a malignancy of the myeloid cell lineage characterized by a recurrent chromosomal abnormality: the Philadelphia chromosome, which results from the reciprocal translocation of the chromosomes 9 and 22. The Philadelphia chromosome contains a fusion gene called BCR-ABL1. The BCR-ABL1 codes for an aberrantly functioning tyrosine kinase that drives the malignant proliferation of the founding clone. The advent of tyrosine kinase inhibitors (TKI) represents a landmark in the treatment of CML, that has led to tremendous improvement in the remission and survival rates. Since the introduction of imatinib, the first TKI, several other TKI have been approved that further broadened the arsenal against CML. Patients treated with TKIs require sensitive monitoring of BCR-ABL1 transcripts with quantitative real-time polymerase chain reaction (qRT-PCT), which has become an essential part of managing patients with CML. In this review, we discuss the importance of the BCR-ABL1 assay, and we highlight the growing importance of BCR-ABL1 dynamics. We also introduce a mathematical correction for the BCR-ABL1 assay that could help homogenizing the use of the ABL1 as a control gene. Finally, we discuss the growing body of evidence concerning treatment-free remission. Along with the continuous improvement in the therapeutic arsenal against CML, the molecular monitoring of CML represents the avant-garde in the struggle to make CML a curable disease.