= Tumour angiogenesis
\b[<Tumour angiogenesis>] is the growth of new <blood vessels> from pre-existing vessels in response to a <tumour> and its surrounding <tissue>. It supplies a route for <oxygen> and nutrients to reach growing <cells>. Vessel formation and effective <perfusion> are distinct: a vascular image alone does not measure the delivery of <oxygen> to every part of a <tumour>.
A foundational experiment by <Judah Folkman>, <Ezio Merler>, <Charles Abernathy> and <Gretchen Williams> tested tumour-derived material in a <rat dorsal air sac assay>. A membrane wick delivered separated material locally; after 48 hours the investigators inspected the vascular response with a stereoscope and took tissue sections. A wick without the active preparation provided a <negative control>. The response was graded from 0 to 5+, rather than measured as an absolute endothelial <cell> count.<Isolation of a Tumor Factor Responsible for Angiogenesis by Judah Folkman and others>[1971, printed p.279, Biological Assay of TAF and Reading the Air Sac Assay; Figures2–4]{ref} This connected an isolated preparation to a reproducible biological effect. It did not identify a unique modern molecule or establish that inhibiting one signal would cure all <cancers>.
An original <diffusion> model explains why access to vessels matters. Approximate a vessel-free <tumour> as a sphere of radius $R$ with uniform <oxygen> consumption $q$ per unit volume per unit time. Let $D$ be a constant <diffusion coefficient> and fix the surface <oxygen> <concentration> at $C_s$. At <steady state>, the <diffusion equation> with consumption and <spherical symmetry> becomes
$$
D\frac{1}{r^2}\frac{\mathrm d}{\mathrm dr}\left(r^2\frac{\mathrm dC}{\mathrm dr}\right)=q.
$$
Finiteness at the centre removes the singular integration term. Integrating again and imposing $C(R)=C_s$ gives
$$
C(r)=C_s-\frac{q}{6D}(R^2-r^2),\qquad C(0)=C_s-\frac{qR^2}{6D}.
$$
If a stipulated threshold $C_{\mathrm{crit}}$ is required to avoid <hypoxia>, the model predicts
$$
R\leq\sqrt{\frac{6D(C_s-C_{\mathrm{crit}})}q}.
$$
For illustration, take $D=2\times10^{-9}\,\mathrm{m^2\,s^{-1}}$, $C_s-C_{\mathrm{crit}}=0.05\,\mathrm{mol\,m^{-3}}$ and $q=0.01\,\mathrm{mol\,m^{-3}\,s^{-1}}$. The limiting radius is approximately $2.45\times10^{-4}\,\mathrm m$, or $0.245\,\mathrm{mm}$. These are chosen model inputs, not measurements from the historical study or a universal size limit. Quadrupling consumption halves the limiting radius. A predicted negative <concentration> indicates failure of the assumed uniformly consuming viable sphere, not physically negative <oxygen>.
The calculation isolates a transport constraint; it does not model vessel branching, changing consumption or delivery by flowing <blood>. Signals such as <vascular endothelial growth factor> can influence <endothelial cells>, but their action and the resulting <perfusion> require biological measurements beyond this spherical model.
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