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CAR T cell therapy in B-cell acute lymphoblastic leukaemia: insights from mathematical models. (English) Zbl 07280108
The authors propose and analyze mathematical models for the response of acute lymphoblastic leukaemias to the injection of CAR T cells. In particular, the following system of ordinary differential equations is considered \begin{align*} \begin{split} \dot C&=\rho_C(L+B)C+\rho_C\beta IC-\frac{1}{\tau_C}C,\\ \dot L&=\rho_LL-\alpha LC,\\ \dot B&=\frac{1}{\tau_I}I -\alpha BC-\frac{1}{\tau_B}B, \\ \dot P&=\rho_P(2a_Ps(P,I)-1)P-\frac{1}{\tau_P}P,\\ \dot I&=\rho_I(2a_Is(P,I)-1)I-\frac{1}{\tau_I}I +\frac{1}{\tau_P}P-\alpha\beta IC, \end{split} \end{align*} where $$C, L,B,P,I$$ are the number of CAR T cells, leukaemic cells, mature healthy B cells, CD19- haematopoietic stem cells and CD19+ B cell progenitors. The signaling function $$s$$ is of the form $$s(P,I) = \frac{1}{1 + k_s ( P + I ) }$$. The first equation describes the proliferation of CAR T cells due to encounters with their target cells and the second equation the proliferation of leukaemic cells and their death due to encounters with CAR T cells. The remaining three equations describe the evolution of B cells which consist of three compartments.
Assuming a steady state approximation for $$I$$ and neglecting the contribution of the haematopoietic compartments, the system is reduced to the simplified model \begin{align*} \begin{split} \dot C&=\rho_C(L+B)C-\frac{1}{\tau_C}C,\\ \dot L&=\rho_LL-\alpha LC,\\ \dot B&= -\alpha BC-\frac{1}{\tau_B}B. \end{split} \end{align*} The dynamics of the reduced system is studied in numerical simulations. In particular, it is shown that the model can explain early post-injection dynamics of the different compartments and the fact that the number of injected CAR T cells does not critically affect the treatment outcome. It also predicts that CD19 + cancer relapses could be the result of competition between leukaemic and CAR T cells.

##### MSC:
 92C50 Medical applications (general) 92C37 Cell biology
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