癌症复发时间的大偏差
Large Deviations of Cancer Recurrence Timing
摘要：
我们研究疾病复发时间的大偏差事件。特别是，我们有兴趣在建模癌症治疗失败，由于突变诱导的耐药性。我们首先提出了这种现象的两种类型的分支过程模型，其中对治疗敏感的细胞的初始群体可以产生对治疗有抗性的突变体。在这个模型中，我们研究了两个随机时间，递归时间和交叉时间。复发时间定义为突变细胞的总体规模首次超过药物敏感细胞初始总体规模的给定比例。交叉时间定义为耐药细胞群体首次占总群体的主导地位。我们建立依概率收敛结果的复发和交叉时间。然后，我们开发表达式的早期复发和早期交叉事件的大偏差率。我们的特点是如何的大偏差率和率函数依赖于突变细胞群体的初始大小。最后，我们看看大偏差率的早期复发条件的突变体克隆的数量，目前在复发的特殊情况下，确定性衰减的敏感人口。我们发现，如果复发发生在预测的大数定律的限制，那么将有可能是在复发时存在的克隆的数量增加。
英文摘要：
We study large deviation events in the timing of disease recurrence. In particular, we are interested in modeling cancer treatment failure due to mutation-induced drug resistance. We first present a two-type branching process model of this phenomenon, where an initial population of cells that are sensitive to therapy can produce mutants that are resistant to the therapy. In this model, we investigate two random times, the recurrence time and the crossover time. Recurrence time is defined as the first time that the population size of mutant cells exceeds a given proportion of the initial population size of drug-sensitive cells. Crossover time is defined as the first time that the resistant cell population dominates the total population. We establish convergence in probability results for both recurrence and crossover time. We then develop expressions for the large deviations rate of early recurrence and early crossover events. We characterize how the large deviation rates and rate functions depend on the initial size of the mutant cell population. We finally look at the large deviations rate of early recurrence conditioned on the number of mutant clones present at recurrence in the special case of a deterministically decaying sensitive population. We find that if recurrence occurs before the predicted law of large numbers limit then there will likely be an increase in the number of clones present at recurrence time.
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