Graduate School of Social Sciences, Tokyo Metropolitan University, Tokyo, Japan.
PLoS Comput Biol. 2019 Feb 19;15(2):e1006770. doi: 10.1371/journal.pcbi.1006770. eCollection 2019 Feb.
The presence of treatment-resistant cells is an important factor that limits the efficacy of cancer therapy, and the prospect of resistance is considered the major cause of the treatment strategy. Several recent studies have employed mathematical models to elucidate the dynamics of generating resistant cancer cells and attempted to predict the probability of emerging resistant cells. The purpose of this paper is to present numerical approach to compute the number of resistant cells and the emerging probability of resistance. Stochastic model was designed and developed a method to approximately but efficiently compute the number of resistant cells and the probability of resistance. To model the progression of cancer, a discrete-state, two-dimensional Markov process whose states are the total number of cells and the number of resistant cells was employed. Then exact analysis and approximate aggregation approaches were proposed to calculate the number of resistant cells and the probability of resistance when the cell population reaches detection size. To confirm the accuracy of computed results of approximation, relative errors between exact analysis and approximation were computed. The numerical values of our approximation method were very close to those of exact analysis calculated in the range of small detection size M = 500, 100, and 1500. Then computer simulation was performed to confirm the accuracy of computed results of approximation when the detection size was M = 10000,30000,50000,100000 and 1000000. All the numerical results of approximation fell between the upper level and the lower level of 95% confidential intervals and our method took less time to compute over a broad range of cell size. The effects of parameter change on emerging probabilities of resistance were also investigated by computed values using approximation method. The results showed that the number of divisions until the cell population reached the detection size is important for emerging the probability of resistance. The next step of numerical approach is to compute the emerging probabilities of resistance under drug administration and with multiple mutation. Another effective approximation would be necessary for the analysis of the latter case.
耐药细胞的存在是限制癌症治疗效果的一个重要因素，而耐药性的出现被认为是治疗策略的主要原因。最近的几项研究利用数学模型阐明了产生耐药癌细胞的动力学，并试图预测耐药细胞出现的概率。本文的目的是提出一种数值方法来计算耐药细胞的数量和耐药出现的概率。设计了随机模型并开发了一种方法，可以近似但有效地计算耐药细胞的数量和耐药出现的概率。为了对癌症的进展进行建模，采用了离散状态的二维马尔可夫过程，其状态是细胞总数和耐药细胞数。然后提出了精确分析和近似聚合方法来计算当细胞群体达到检测大小时耐药细胞的数量和耐药出现的概率。为了确认近似计算结果的准确性，计算了精确分析和近似之间的相对误差。在小检测大小 M = 500、100 和 1500 的范围内，我们的近似方法的数值非常接近精确分析的计算值。然后进行计算机模拟，以确认当检测大小为 M = 10000、30000、50000、100000 和 1000000 时近似计算结果的准确性。近似方法的所有数值结果都落在 95%置信区间的上限和下限之间，并且我们的方法在广泛的细胞大小范围内计算时间更短。还通过近似方法计算的数值研究了参数变化对耐药出现概率的影响。结果表明，细胞群体达到检测大小所需的分裂次数对于耐药出现的概率很重要。数值方法的下一步是在药物治疗和多种突变下计算耐药出现的概率。在后一种情况下，需要另一种有效的近似方法来进行分析。
