Department of Aerospace & Mechanical Engineering and Department of Mathematics, University of Southern California, Los Angeles, California, United States of America.
PLoS One. 2012;7(4):e34637. doi: 10.1371/journal.pone.0034637. Epub 2012 Apr 27.
A stochastic Markov chain model for metastatic progression is developed for primary lung cancer based on a network construction of metastatic sites with dynamics modeled as an ensemble of random walkers on the network. We calculate a transition matrix, with entries (transition probabilities) interpreted as random variables, and use it to construct a circular bi-directional network of primary and metastatic locations based on postmortem tissue analysis of 3827 autopsies on untreated patients documenting all primary tumor locations and metastatic sites from this population. The resulting 50 potential metastatic sites are connected by directed edges with distributed weightings, where the site connections and weightings are obtained by calculating the entries of an ensemble of transition matrices so that the steady-state distribution obtained from the long-time limit of the Markov chain dynamical system corresponds to the ensemble metastatic distribution obtained from the autopsy data set. We condition our search for a transition matrix on an initial distribution of metastatic tumors obtained from the data set. Through an iterative numerical search procedure, we adjust the entries of a sequence of approximations until a transition matrix with the correct steady-state is found (up to a numerical threshold). Since this constrained linear optimization problem is underdetermined, we characterize the statistical variance of the ensemble of transition matrices calculated using the means and variances of their singular value distributions as a diagnostic tool. We interpret the ensemble averaged transition probabilities as (approximately) normally distributed random variables. The model allows us to simulate and quantify disease progression pathways and timescales of progression from the lung position to other sites and we highlight several key findings based on the model.
基于转移部位的网络构建，我们为原发性肺癌开发了一种转移性进展的随机马尔可夫链模型，其中转移的动力学模拟为网络上的随机游动者集合。我们计算了一个转移矩阵，其条目（转移概率）被解释为随机变量，并使用它来构建基于 3827 例未经治疗的患者死后组织分析的原发性和转移性部位的循环双向网络，该分析记录了该人群中所有原发性肿瘤部位和转移性部位。由分布权重连接的 50 个潜在转移部位通过有向边缘连接，其中站点连接和权重是通过计算转移矩阵集合的条目获得的，以便从马尔可夫链动力系统的长时间极限获得的稳态分布对应于从尸检数据集获得的集合转移分布。我们将转移矩阵的搜索条件限制在从数据集获得的转移性肿瘤的初始分布上。通过迭代数值搜索过程，我们调整一系列逼近的条目，直到找到具有正确稳态的转移矩阵（达到数值阈值）。由于这个受约束的线性优化问题是不确定的，我们将使用奇异值分布的平均值和方差来计算转移矩阵集合的统计方差作为诊断工具。我们将集合平均转移概率解释为（近似）正态分布的随机变量。该模型允许我们模拟和量化疾病进展途径以及从肺部到其他部位的进展时间尺度，并且我们基于该模型强调了几个关键发现。
