Based upon the deterministic Gompertz law of cell growth, we have proposed a stochastic model of tumour cell growth, in which the size of the tumour cells is bounded. The model takes account of both cell fission (which is an ‘action at a distance’ effect) and mortality too. Accordingly, the density function of the size of the tumour cells obeys a functional Fokker–Planck Equation (FPE) associated with the bounded stochastic process. We apply the Lie-algebraic method to derive the exact analytical solution via an iterative approach. It is found that the density function exhibits an interesting kink-like structure generated by cell fission as time evolves.

中文翻译：

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肿瘤细胞生长的改良随机Gompertz模型

基于细胞生长的确定性Gompertz定律，我们提出了肿瘤细胞生长的随机模型，其中肿瘤细胞的大小是有界的。该模型同时考虑了细胞分裂（这是“远距离作用”效应）和死亡率。因此，肿瘤细胞大小的密度函数服从与有界随机过程相关的函数Fokker-Planck方程（FPE）。我们采用李-代数方法通过迭代方法得出精确的解析解。发现随着时间的发展，密度函数表现出由细胞裂变产生的有趣的扭结状结构。