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Chimeric Antigen Receptor (CAR) T-cell therapy has demonstrated high rates of response in recurrent B-cell Acute Lymphoblastic Leukemia in children and young adults. Despite this success, a fraction of patients' experience relapse after treatment. Relapse is often preceded by recovery of healthy B cells, which suggests loss or dysfunction of CAR T-cells in bone marrow. This site is harder to access, and thus is not monitored as frequently as peripheral blood. Understanding the interplay between B cells, leukemic cells, and CAR T-cells in bone marrow is paramount in ascertaining the causes of lack of response. In this paper, we put forward a mathematical model representing the interaction between constantly renewing B cells, CAR T-cells, and leukemic cells in the bone marrow. Our model accounts for the maturation dynamics of B cells and incorporates effector and memory CAR T-cells. The model provides a plausible description of the dynamics of the various cellular compartments in bone marrow after CAR T infusion. After exploration of the parameter space, we found that the dynamics of CAR T product and disease were independent of the dose injected, initial B-cell load, and leukemia burden. We also show theoretically the importance of CAR T product attributes in determining therapy outcome, and have studied a variety of possible response scenarios, including second dosage schemes. We conclude by setting out ideas for the refinement of the model., This work was partially supported by the Fundacion Espanola para la Ciencia y la Tecnologia (UCA PR214), the Asociacion Pablo Ugarte (APU, Spain), Junta de Comunidades de Castilla-La Mancha (SBPLY/17/180501/000154), Ministry of Science and Technology, Spain (PID2019110895RB-I00), and Inversion Territorial Integrada de la Provincia de Cadiz (ITI-0038-2019).

Haematopoiesis is the process of generation of blood cells. Lymphopoiesis generates lymphocytes, the cells in charge of the adaptive immune response. Disruptions of this process are associated with diseases like leukaemia, which is especially incident in children. The characteristics of self-regulation of this process make them suitable for a mathematical study. In this paper we develop mathematical models of lymphopoiesis using currently available data. We do this by drawing inspiration from existing structured models of cell lineage development and integrating them with paediatric bone marrow data, with special focus on regulatory mechanisms. A formal analysis of the models is carried out, giving steady states and their stability conditions. We use this analysis to obtain biologically relevant regions of the parameter space and to understand the dynamical behaviour of B-cell renovation. Finally, we use numerical simulations to obtain further insight into the influence of proliferation and maturation rates on the reconstitution of the cells in the B line. We conclude that a model including feedback regulation of cell proliferation represents a biologically plausible depiction for B-cell reconstitution in bone marrow. Research into haematological disorders could benefit from a precise dynamical description of B lymphopoiesis., Comment: Submitted to Journal of Theoretical Biology

Immunotherapies use components of the patient immune system to selectively target cancer cells. The use of CAR T cells to treat B-cell malignancies – leukaemias and lymphomas– is one of the most successful examples, with many patients experiencing long-lasting complete responses to this therapy. This treatment works by extracting the patient’s T cells and adding them the CAR group, which enables them to recognize and target cells carrying the antigen CD19+, that is expressed in these haematological tumors.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, tumor cells, CAR T cells and side effects by incorporating 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 provides 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+ tumor relapses could be the result of the competition between tumor and CAR T cells analogous to predator-prey dynamics. We discuss this fact on the light of available evidences and the possibility of controlling relapses by early re-challenging of the tumor with stored CAR T cells.