多类型分支过程种群大小分布的数值近似

中文摘要 AI 辅助

连续时间多类型分支过程是描述种群扩张和迁移的基础模型，癌症演化是其典型示例。从时间序列计数数据推断模型参数（如突变率和生长率）需要高效计算种群大小分布。现有方法主要基于长时间或大数量渐近，依赖于受限的初始条件或简化细胞类型间相互作用。在此，我们引入两种针对有向图上具有任意初始化的多类型分支过程种群大小分布的数值近似方法。第一种方法将鞍点近似与概率生成函数的数值积分相结合。我们刻画了可容许性，并建立了鞍点存在性和唯一性的条件。对于有向无环图，第二种方法基于闭式近似拉普拉斯变换和高效数值反演，提供了大时间小突变率的替代方案。我们在模拟中基准测试了两种解的精度和速度，显示出相对于最先进的大数量近似有显著改进，并在匹配精度下比Gillespie随机模拟算法快数个数量级。我们将方法应用于分析一名急性髓系白血病患者的复发动态，其中对六类型患者特异性突变树的快速参数扫描量化了未观察到的缓解期负担和治疗改变的适应性如何解释复发。我们的方法为未来癌症演化及其他扩张种群中基于似然的推断提供了计算构建模块。

英文摘要

Continuous-time multi-type branching processes are fundamental models for expanding and migrating populations with cancer evolution being a prototypical example. Inferring model parameters, like mutation and growth rates, from time-series count data requires efficient computation of population size distributions. Existing methods are mainly based on large-time or large-number asymptotics, which rely on either restricted initial conditions or simplified interactions between cell types. Here, we introduce two numerical approximations of population size distributions for multi-type branching processes on directed graphs with arbitrary initialization. The first approach combines a saddle-point approximation with numerical integration of the probability generating function. We characterize admissibility and establish conditions for saddle-point existence and uniqueness. For directed acyclic graphs, the second approach provides a large-time small-mutation-rate alternative based on closed-form approximate Laplace transforms and efficient numerical inversion. We benchmark the accuracy and speed of both solutions in simulations, showing substantial improvement over the state-of-the-art large-number approximation and orders of magnitude speedup over Gillespie's stochastic simulation algorithm at matching accuracy. We apply our methods to analyze the relapse dynamics of an acute myeloid leukemia patient, where rapid parameter scans over a six-type patient-specific mutation tree quantify how unobserved remission burden and treatment-altered fitness can explain relapse. Our methods provide computational building blocks for future likelihood-based inference in cancer evolution and other expanding populations.

发表机构

• ETH Zurich（苏黎世联邦理工学院）
• SIB Swiss Institute of Bioinformatics（瑞士生物信息学研究所）

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