Abstract
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.
Highlights
Oncogenic pathways have been investigated using molecular biology techniques, which have helped elucidate the molecular mechanism of cancer growth, invasion, and metastasis among other aspects
Drug therapies for cancer have dramatically succeeded since molecular-targeted drugs have been introduced in medical practice; drug treatment often fails owing to the emergence of drug-resistant cells
This paper proposes one of the mathematical approaches that uses a stochastic model and provides the emerging probabilities of resistance at detection size
Summary
Oncogenic pathways have been investigated using molecular biology techniques, which have helped elucidate the molecular mechanism of cancer growth, invasion, and metastasis among other aspects. The findings from these investigations have encouraged the development of anti-cancer drugs that inhibit specific oncogenic pathway and have helped improve clinical outcomes dramatically [1] [2]. Mathematical models are often used to elucidate the dynamics of evolutionary process and have been studied to understand the mechanism through which cancer cells develop drug resistance [4] [6] [7] [8] [9] [10]
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