CAR T cell therapy in B-cell acute lymphoblastic leukaemia: Insights from mathematical models

•A mathematical model of leukaemia response to CAR-T cell treatment is built and studied.•Tumour relapse is a dynamical phenomenon and can be controlled.•The mathematical model allows obtaining a formula for the relapse time.•Persistence of the disease depends on the generation of B-cells in the bone marrow. Immunotherapies use components of the patient immune system to selectively target cancer cells. The use of chimeric antigenic receptor (CAR) T cells to treat B-cell malignancies –leukaemias and lymphomas– is one of the most successful examples, with many patients experiencing long-lasting full responses to this therapy. This treatment works by extracting the patient’s T cells and transducing them with the CAR, enabling them to recognize and target cells carrying the antigen CD19+, which is expressed in these haematological cancers. Here we put forward a mathematical model describing the time response of leukaemias to the injection of CAR T cells. The model accounts for mature and progenitor B-cells, leukaemic cells, CAR T cells and side effects by including the main biological processes involved. The model explains the early post-injection dynamics of the different compartments and the fact that the number of CAR T cells injected does not critically affect the treatment outcome. An explicit formula is found that gives the maximum CAR T cell expansion in vivo and the severity of side effects. Our mathematical model captures other known features of the response to this immunotherapy. It also predicts that CD19+ cancer relapses could be the result of competition between leukaemic and CAR T cells, analogous to predator-prey dynamics. We discuss this in the light of the available evidence and the possibility of controlling relapses by early re-challenging of the leukaemia cells with stored CAR T cells.