Interplay between distribution of live cells and growth dynamics of solid tumours

Abstract

Experiments show that simple diffusion of nutrients and waste molecules is not sufficient to explain the typical multilayered structure of solid tumours, where an outer rim of proliferating cells surrounds a layer of quiescent but viable cells and a central necrotic region. These experiments challenge models of tumour growth based exclusively on diffusion. Here we propose a model of tumour growth that incorporates the volume dynamics and the distribution of cells within the viable cell rim. The model is suggested by in silico experiments and is validated using in vitro data. The results correlate with in vivo data as well, and the model can be used to support experimental and clinical oncology.

Pulsation-limited oxygen diffusion in the tumour microenvironment

“…2). To take this structure into account, we assume that the viable cell density decreases roughly exponentially with increasing distance from the nutrient supply system as discussed in refs 24 and 25 and that the local oxygen consumption is proportional to cell density. This means that the consumption rate γ is a decreasing function of the radius r …”

“…All the parameters used in the numerical evaluation are extrapolated from experimental data and apply to solid tumours. We take the decay length of the exponential reduction of the consumption rate λ c  = 120 μm from refs 24 and 25, and the diffusion constant of oxygen as measured both in blood and tissues2627) D  = 2 × 10 −9  m 2 /s. The rates of oxygen consumption in different areas of in vivo tumours have been elegantly and precisely measured, and they have been shown to vary in the range 1.66 10 −4 –5 10 −3  s −1 (mean value 2.16 10 −3  s −1 )282930.…”

Metabolic scaling in solid tumours

“…Recently, by combining the diffusion of oxygen, nutrients, and metabolites in the extracellular environment, and the internal motions that mix live and dead cells, we derived a growth law of solid tumours which is linked to parameters at the cellular level1. Here we use this growth law to obtain a metabolic scaling law for solid tumours, which is obeyed by tumours of different histotypes both in vitro and in vivo , and we display its relation with the fractal dimension of the distribution of live cells in the tumour mass.…”

“…Recently we derived a growth law for solid tumours1, where growth depends on the distribution of live cells in the tumour mass. The model was suggested by the results of computer simulations2, and it has been validated using data from in vitro experiments.…”

“…We start from the tumour growth law, that can be expressed as a differential equation for tumour volume, and combines proliferation of live cells in the tumour with the gradual shrinking of dead cells1 where is the fraction of live cells in the tumour, derived from the assumption of exponential decay of the density of live cells, so that the total volume of live cells is V a = F ( V ) V , and where the parameter α defines the individual cell proliferation rate, while the parameter δ is the shrinking rate of dead cells. The tumour volume is proportional to x 3 , where x is some characteristic length of the tumour, i.e., V = Ax 3 .…”