Oxygen Diffusion in Tissue — Krogh's Cylinder Model
Every mitochondrion in your body sits within a few tens of micrometres of a capillary — not by accident, but because oxygen molecules have no engine of their own. They move by pure passive diffusion, a random walk down a concentration gradient, and diffusion is ferociously slow over anything but microscopic distances. August Krogh's 1919 cylinder model, built from measurements of capillary density in resting and working muscle, turned this constraint into a quantitative theory that still underlies how we understand hypoxia, tumour biology, exercise physiology and the engineering limits of artificial tissue.
1. Fick's First Law of Diffusion
Adolf Fick described diffusive transport in 1855: the flux of a solute is proportional to the local concentration gradient. For oxygen crossing a thin slab of tissue:
The minus sign says flux runs from high to low concentration — O₂ always flows from the oxygen-rich capillary lumen (PO₂ ≈ 100 mmHg arterial) toward the oxygen-poor mitochondria (PO₂ as low as 1–3 mmHg at the point of use), never the reverse, exactly as required by the second law of thermodynamics.
2. Why Diffusion Is Fast at Micron Scale, Useless at Centimetre Scale
The time for a diffusing particle to travel a distance L scales with L², not L — this quadratic scaling is the entire reason multicellular organisms need circulatory systems at all:
A tissue block more than roughly half a millimetre from its nearest capillary cannot be kept oxygenated by diffusion alone at typical metabolic rates — this single number, not any exotic biology, fixes capillary density at 1 vessel roughly every 20–50 μm in most tissues, and is the central design constraint that convection (blood flow) exists to bypass: circulation delivers O₂ over centimetres to metres at bulk-flow speed, then diffusion finishes the job over the last tens of micrometres.
3. The Krogh Cylinder Model
August Krogh (1874–1949, Nobel Prize in Physiology or Medicine 1920) modelled tissue as concentric cylinders of tissue surrounding each capillary, each supplying O₂ to a fixed radius r_t before the next capillary's territory begins:
The product D·α is called the Krogh diffusion coefficient, K = D·α, and is the single empirical constant Krogh measured directly by timing bubble absorption through gelatin and muscle — it collapses two unknowns (how fast O₂ moves, how much dissolves) into one measurable number.
4. Oxygen Solubility and the Krogh Diffusion Coefficient
Diffusion coefficient D
O₂ in water/tissue at 37 °C: D ≈ 1.7–2.4 × 10⁻⁵ cm²/s. About 5–10× slower than in air (D_air ≈ 2 × 10⁻¹ cm²/s) because liquid water is far more viscous than gas.
Solubility α (Bunsen)
O₂ dissolves poorly in water: α ≈ 2.4 × 10⁻⁵ mL O₂/(mL·mmHg) at 37 °C — Henry's law, C_dissolved = α · P. This low solubility is why oxygen alone in plasma cannot meet metabolic demand.
Krogh's K = D·α
K ≈ 1.4 × 10⁻¹⁴ mL O₂ / (cm·s·mmHg) in resting skeletal muscle — Krogh's original 1919 measurement, still cited in modern tumour-oxygenation models a century later.
Myoglobin facilitation
Muscle myoglobin acts as an intracellular O₂ shuttle, effectively boosting apparent D by 1.5–2× at low PO₂ by "facilitated diffusion" — molecules hop from one myoglobin site to the next.
5. Haemoglobin: the Oxygen Buffer
Free dissolved O₂ in plasma cannot supply the body's demand — a litre of blood carries only ≈ 3 mL of dissolved O₂ at arterial PO₂, versus ≈ 200 mL bound to haemoglobin. The sigmoidal oxygen–haemoglobin dissociation curve is what makes this buffer work efficiently across the whole PO₂ range the body encounters:
The steep middle portion of the sigmoid (roughly 20–60 mmHg) is exactly where the capillary-to-tissue PO₂ drop happens, so small changes in tissue PO₂ release comparatively large amounts of O₂ — the curve's shape is itself an adaptation that keeps the driving gradient for Fick diffusion high along the whole capillary length, rather than collapsing to zero early.
6. The Lethal Corner and Tissue Hypoxia
Combining the Krogh–Erlang profile with the falling PO₂ along the capillary length (as O₂ is extracted from venous to arterial end) produces a two-dimensional map of tissue oxygenation. The lowest PO₂ in the whole cylinder sits at the venous end, at the farthest radius from the capillary — Krogh's "lethal corner":
Physiological responses to the diffusion limit:
- Capillary recruitment: resting muscle perfuses only a fraction of its capillary bed; exercise opens dormant capillaries, cutting r_t and raising minimum tissue PO₂.
- Angiogenesis (VEGF pathway): chronic hypoxia (HIF-1α stabilisation) triggers new capillary growth, permanently shrinking the diffusion distance — the same pathway tumours hijack to escape diffusion limits.
- Myoglobin buffering: stores O₂ intracellularly and smooths transient supply/demand mismatches during contraction cycles.
- Erythropoiesis: chronic hypoxia (altitude, anaemia) raises haematocrit, increasing O₂ carrying capacity per litre of blood rather than diffusion distance itself.
7. JavaScript Krogh Cylinder Simulator
// Krogh cylinder: steady-state radial PO2 profile around a capillary
// P(r) = Pc - (M / (4*D*alpha)) * [r^2 - rc^2 - 2*rt^2*ln(r/rc)]
function kroghProfile({
Pc = 40, // capillary wall PO2 [mmHg]
rc = 3e-4, // capillary radius [cm] (~3 um)
rt = 2.5e-3, // tissue cylinder radius [cm] (~25 um)
M = 2e-4, // O2 consumption [mL O2 / (mL tissue . s)]
D = 1.7e-5, // diffusion coeff [cm^2/s]
alpha = 2.4e-5 // solubility [mL O2 / (mL . mmHg)]
} = {}, samples = 40) {
const K = D * alpha; // Krogh diffusion coefficient
const profile = [];
for (let i = 0; i <= samples; i++) {
const r = rc + (rt - rc) * (i / samples);
const term = r**2 - rc**2 - 2 * rt**2 * Math.log(r / rc);
const P = Pc - (M / (4 * K)) * term;
profile.push({ r_um: (r * 1e4).toFixed(1), P_mmHg: Math.max(0, P).toFixed(2) });
}
return profile;
}
// Maximum sustainable tissue radius: find rt where P(rt) just reaches 0
function maxRadius(Pc, rc, M, K, tolerance = 1e-6) {
let lo = rc, hi = rc * 50;
function Pboundary(rt) {
const term = rt**2 - rc**2 - 2 * rt**2 * Math.log(rt / rc);
return Pc - (M / (4 * K)) * term;
}
while (hi - lo > tolerance) {
const mid = (lo + hi) / 2;
if (Pboundary(mid) > 0) lo = mid; else hi = mid;
}
return lo; // [cm]
}
const profile = kroghProfile({});
const minP = Math.min(...profile.map(p => +p.P_mmHg));
console.log(`Minimum tissue PO2 (lethal corner): ${minP.toFixed(1)} mmHg`);
const K = 1.7e-5 * 2.4e-5;
const rMax = maxRadius(40, 3e-4, 2e-4, K);
console.log(`Critical radius before anoxia: ${(rMax * 1e4).toFixed(1)} um`);
8. Physiological and Clinical Consequences
Exercise physiology
During maximal exertion M rises ~20×; the body compensates by capillary recruitment and dilation, not by increasing D or α (both are fixed tissue properties), which is why training increases capillary density (angiogenesis) rather than diffusivity.
Tumour hypoxia
Tumours often outgrow their vascular supply, creating hypoxic cores beyond ~150 μm from the nearest vessel — these radioresistant regions (low O₂ blunts radiotherapy's free-radical mechanism) drive anti-angiogenic and hypoxia-activated-prodrug treatment strategies.
Wound healing
Ischaemic wounds fail to heal partly because damaged microvasculature increases effective r_t beyond the diffusion limit; hyperbaric oxygen therapy raises P_c (arterial PO₂ up to ~2000 mmHg at 3 atm), extending the viable radius.
Tissue engineering
Lab-grown organoids and thick engineered tissue constructs (>~200 μm) necrose at the core without a built-in vascular network — the same Krogh-radius constraint that shapes real organs limits how large a scaffold can be before it needs perfusable channels.