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.

免疫疗法使用患者免疫系统的成分选择性地靶向癌细胞。嵌合抗原受体（CAR）T细胞用于治疗B细胞恶性肿瘤（白血病和淋巴瘤）是最成功的例子之一，许多患者对此疗法产生了长期的全面反应。这种治疗方法是通过提取患者的T细胞并用CAR转导它们，使其能够识别和靶向携带CD19抗原的细胞而起作用${}^{+}，$在这些血液学癌症中表达。

在这里，我们提出了描述白血病对注射CAR T细胞的时间响应的数学模型。该模型通过包括所涉及的主要生物学过程，解释了成熟和祖细胞B细胞，白血病细胞，CAR T细胞和副作用。该模型解释了不同隔室的早期注射后动态，以及注射的CAR T细胞数量不会严重影响治疗结果这一事实。发现了一个明确的公式，该公式可在体内提供最大的CAR T细胞扩增，并具有严重的副作用。我们的数学模型捕获了对该免疫疗法反应的其他已知特征。它还预测CD19${}^{+}$癌症复发可能是白血病细胞和CAR T细胞竞争的结果，类似于捕食者-猎物的动态。我们根据现有证据和通过与存储的CAR T细胞早期重新挑战白血病细胞来控制复发的可能性来讨论这一点。