The detection, in a wide variety of cancer types, of a population of highly tumorigenic cells that exhibit self-renewal and multipotency, which are hallmarks of stem cells, has transformed the current view of tumor initiation, progression, and treatment. Here, we develop and analyze a mathematical model for tumor growth that is based on the current biological understanding of the processes that underlie cellular expansion under the hierarchical guidelines of the cancer stem cell (CSC) hypothesis. Important features of the model include (i) a nonlinear probability of CSC self-renewal that reflects the fact that this key type of stem cell division can be regulated by extrinsic and intrinsic chemical signaling as well as environmental (niche) constraints and (ii) an amplification factor that captures the transient amplifying divisions that are a defining characteristic of progenitor cells. We present a thorough mathematical analysis of the model and highlight the conditions required for tumors to evolve toward either bounded or exponential growth. Numerical simulations further illustrate the impact of the various parameters on the tumor growth rate and on the heterogeneous cellular composition, which varies during progression.

中文翻译：

---

具有非线性自我更新的癌症干细胞驱动肿瘤生长模型中异质性的数学表征

在多种癌症类型中检测到具有自我更新和多能性的高致瘤细胞群，这是干细胞的标志，改变了目前对肿瘤发生、进展和治疗的看法。在这里，我们开发和分析了一个肿瘤生长的数学模型，该模型基于当前对癌症干细胞 (CSC) 假设的分级准则下细胞扩增过程的生物学理解。该模型的重要特征包括 (i) CSC 自我更新的非线性概率，这反映了这种关键类型的干细胞分裂可以通过外在和内在的化学信号以及环境（利基）约束来调节，以及 (ii)一种放大因子，它捕获瞬时放大分裂，这是祖细胞的定义特征。我们对该模型进行了全面的数学分析，并强调了肿瘤向有界或指数增长发展所需的条件。数值模拟进一步说明了各种参数对肿瘤生长速率和异质细胞组成的影响，这些组成在进展过程中会发生变化。我们对该模型进行了全面的数学分析，并强调了肿瘤向有界或指数增长发展所需的条件。数值模拟进一步说明了各种参数对肿瘤生长速率和异质细胞组成的影响，这些组成在进展过程中会发生变化。我们对该模型进行了全面的数学分析，并强调了肿瘤向有界或指数增长发展所需的条件。数值模拟进一步说明了各种参数对肿瘤生长速率和异质细胞组成的影响，这些组成在进展过程中会发生变化。