Cancer cells grown in culture can form a solid ball
PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Physical foundations of the living cell · Physical foundations of the living cell
Life at Thermal Energy, Ch Random walks, diffusion and the crowded cell
Problem
Cancer cells grown in culture can form a solid ball, a tumour spheroid, fed with oxygen only by diffusion from the medium around it. Take the oxygen concentration at its surface as c0=0.20molm−3, the diffusion coefficient of oxygen inside it as D=2×10−9m2s−1, and a uniform consumption rate Γ=2×10−2molm−3s−1 throughout the living tissue.
((a)) Adapt the chapter's slab estimate of how thick a tissue diffusion alone can supply to a sphere of radius R fed over its whole surface, and find the largest radius whose centre is still supplied.
((b)) Evaluate that radius and the corresponding diameter.
((c)) Spheroids grown larger than about half a millimetre across are commonly found to have a dead core. Compare this with your answer, and say how closely an estimate of this kind should be expected to agree.
((d)) Does a spheroid twice as large as your answer die throughout, or only in part? Estimate the thickness of whatever stays alive.