Immunotherapy has the potential to change the way all cancer types are
treated and cured. Cancer immunotherapies use elements of the patient immune
system to attack tumor cells. One of the most successful types of immunotherapy
is CAR-T cells. This treatment works by extracting patient's T-cells and adding
to them an antigen receptor allowing tumor cells to be recognized and targeted.
These new cells are called CAR-T cells and are re-infused back into the patient
after expansion in-vitro. This approach has been successfully used to treat
B-cell malignancies (B-cell leukemias and lymphomas). However, its application
to the treatment of T-cell leukemias faces several problems. One of these is
fratricide, since the CAR-T cells target both tumor and other CAR-T cells. This
leads to nonlinear dynamical phenomena amenable to mathematical modeling. In
this paper we construct a mathematical model describing the competition of
CAR-T, tumor and normal T-cells and studied some basic properties of the model
and its practical implications. Specifically, we found that the model
reproduced the observed difficulties for in-vitro expansion of the therapeutic
cells found in the laboratory. The mathematical model predicted that CAR-T cell
expansion in the patient would be possible due to the initial presence of a
large number of targets. We also show that, in the context of our mathematical
approach, CAR-T cells could control tumor growth but not eradicate the disease