癌症是一种复杂而致命的健康问题，其特征是形成潜在恶性肿瘤或癌细胞。这些细胞与其环境之间的动态相互作用对疾病至关重要。数学模型可以增进我们对这些相互作用的理解，帮助我们预测疾病进展和治疗策略。在本研究中，我们为肺癌特别开发了一个分数肿瘤-免疫相互作用模型（FTIIM-LC）。我们介绍了与Caputo算子相关的一些定义和重要结果。我们采用广义拉盖尔多项式（GLPs）方法找到了FTIIM-LC模型的最优解。然后，我们进行了数值模拟，并将我们的方法的结果与其他技术和真实数据进行了比较。在本文中，我们提出了一个FTIIM-LC模型。所提出模型的近似解是利用一组新的多项式，即GLPs，进行扩展得到的。为了简化过程，我们整合了拉格朗日乘子、GLPs以及分数和普通导数的运算矩阵。我们进行了数值模拟，研究了不同分数阶的影响，并达到了预期的理论结果。本研究的发现表明，所使用的优化方法能够有效地预测和分析复杂现象。这种创新方法还可以应用于其他非线性微分方程，如分数Klein-Gordon方程、分数扩散-波动方程、乳腺癌模型和分数最优控制问题。© 2023. BioMed Central Ltd., Springer Nature的一部分。

Cancer, a complex and deadly health concern today, is characterized by forming potentially malignant tumors or cancer cells. The dynamic interaction between these cells and their environment is crucial to the disease. Mathematical models can enhance our understanding of these interactions, helping us predict disease progression and treatment strategies.In this study, we develop a fractional tumor-immune interaction model specifically for lung cancer (FTIIM-LC). We present some definitions and significant results related to the Caputo operator. We employ the generalized Laguerre polynomials (GLPs) method to find the optimal solution for the FTIIM-LC model. We then conduct a numerical simulation and compare the results of our method with other techniques and real-world data.We propose a FTIIM-LC model in this paper. The approximate solution for the proposed model is derived using a series of expansions in a new set of polynomials, the GLPs. To streamline the process, we integrate Lagrange multipliers, GLPs, and operational matrices of fractional and ordinary derivatives. We conduct a numerical simulation to study the effects of varying fractional orders and achieve the expected theoretical results.The findings of this study demonstrate that the optimization methods used can effectively predict and analyze complex phenomena. This innovative approach can also be applied to other nonlinear differential equations, such as the fractional Klein-Gordon equation, fractional diffusion-wave equation, breast cancer model, and fractional optimal control problems.© 2023. BioMed Central Ltd., part of Springer Nature.