Generalized Lerch polynomials: application in fractional model of CAR-T cells for T-cell leukemia

Chimeric antigen receptor T-cell (CAR-T) therapy has emerged as a widely accepted approach in the management of blood malignancies, enabling specifically trained CAR-T cells to target cancerous cells. Encouraging outcomes have been observed in B-cell lymphoma patients who have exhibited complete responses to CAR-T cell therapy. Ongoing researches are exploring the therapeutic effects of CAR-T cells on various tumors and malignancies, including T cell leukemia. The aim of this study is to present an optimized method for solving the fractional model of CAR-T cell treatment for patients with T-cell leukemia by means of new basis functions, generalized Lerch polynomials (GLPs). The optimization method is performed through the Lagrange multipliers. To observe the correctness, efficiency, competence, and proficiency of the designed computing algorithm, an exhaustive analysis of different parameters, CPU times, and residual function optimal values are presented. Our results predicted a significant decrease in the count of leukemic T-cells after the first dose administration of CAR-T cells to T leukemia patients. Additionally, our mathematical analysis demonstrated that an optimized cure may occur after the second injection of CAR-T cells. These findings underscore the efficacy and potential of our optimized method using GLPs for solving the fractional model of CAR-T cell treatment in patients with T-cell leukemia.