Breward C J W, Byrne H M, Lewis C E
Centre for Mathematical Medicine, Division of Theoretical Mechanics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
J Math Biol. 2002 Aug;45(2):125-52. doi: 10.1007/s002850200149.
A two-phase model is presented to describe avascular tumour growth. Conservation of mass equations, including oxygen-dependent cell growth and death terms, are coupled with equations of momentum conservation. The cellular phase behaves as a viscous liquid, while the viscosity of the extracellular water manifests itself as an interphase drag. It is assumed that the cells become mechanically stressed if they are too densely packed and that the tumour will try to increase its volume in order to relieve such stress. By contrast, the overlapping filopodia of sparsely populated cells create short-range attractive effects. Finally, oxygen is consumed by the cells as it diffuses through the tumour. The resulting system of equations are reduced to three, which describe the evolution of the tumour cell volume fraction, the cell speed and the oxygen tension. Numerical simulations indicate that the tumour either evolves to a travelling wave profile, in which it expands at a constant rate, or it settles to a steady state, in which the net rates of cell proliferation and death balance. The impact of varying key model parameters such as cellular viscosity, interphase drag, and cellular tension are discussed. For example, tumours consisting of well-differentiated (i.e. viscous) cells are shown to grow more slowly than those consisting of poorly-differentiated (i.e. less viscous) cells. Analytical results for the case of oxygen-independent growth are also presented, and the effects of varying the key parameters determined (the results are in line with the numerical simulations of the full problem). The key results and their biological implications are then summarised and future model refinements discussed.
提出了一个两阶段模型来描述无血管肿瘤的生长。质量守恒方程，包括依赖氧气的细胞生长和死亡项，与动量守恒方程相耦合。细胞相表现为粘性液体，而细胞外水的粘性表现为相间阻力。假设如果细胞堆积过于密集，它们会受到机械应力，并且肿瘤会试图增加其体积以减轻这种应力。相比之下，稀疏分布的细胞的重叠丝状伪足会产生短程吸引作用。最后，氧气在扩散通过肿瘤时被细胞消耗。由此产生的方程组被简化为三个，用于描述肿瘤细胞体积分数、细胞速度和氧张力的演变。数值模拟表明，肿瘤要么演变为行波剖面，以恒定速率扩张，要么稳定到稳态，此时细胞增殖和死亡的净速率达到平衡。讨论了改变关键模型参数（如细胞粘度、相间阻力和细胞张力）的影响。例如，由高分化（即粘性大）细胞组成的肿瘤比由低分化（即粘性小）细胞组成的肿瘤生长得更慢。还给出了与氧气无关生长情况下的解析结果，并确定了改变关键参数的影响（结果与完整问题的数值模拟一致）。然后总结了关键结果及其生物学意义，并讨论了未来模型的改进。
