HPC Tier Module
Krogh Cylinder Oxygen Transport
Solve radial tissue oxygenation against the exact Krogh-Erlang profile and locate the lethal corner.
See it run - a worked example, 100% in this browser tab
The problem
Tissue-oxygenation teaching and model exploration need a Krogh solver that reports the lethal-corner pO2 and any anoxic radius honestly, validated against an exact profile rather than asserted.
The local-first solution
This plugin solves the cited Krogh-Erlang radial BVP by finite differences in exact f64, compares it to the closed-form pO2 profile, and reports the critical consumption and lethal radius by bisection - all in your browser with no upload or API key.
What it does
Radial finite-difference BVP with ghost-node Neumann no-flux and Thomas tridiagonal solve
Exact Krogh-Erlang closed-form pO2 profile as the validation benchmark
Reports the L-infinity error between the numeric solve and the closed form
Critical consumption M* at which pO2 reaches zero before the outer radius
Lethal radius (anoxic boundary) found by bisection when consumption exceeds M*
Optional Michaelis-Menten saturating consumption checked in its zero-order limit
Honest scope
Exact: the zero-order Krogh-Erlang closed form, the radial finite-difference BVP, the Thomas solve, the critical-consumption formula, and the bisection lethal radius; the Michaelis-Menten option has no closed form and is validated only in its zero-order limit. It assumes cylindrical symmetry, steady state, a single isolated Krogh cylinder, and constant diffusivity and solubility (all surfaced inputs), and does not model axial gradients, hemoglobin or myoglobin kinetics, inter-capillary competition, or any 2D/3D geometry. Research and education only - not for clinical, diagnostic, or treatment decisions.
Authorities cited
- Krogh, A. (1919). The number and distribution of capillaries in muscles with calculations of the oxygen pressure head necessary for supplying the tissue. The Journal of Physiology 52(6), 409-415. DOI 10.1113/jphysiol.1919.sp001839. - The original tissue-cylinder model and the "lethal corner" concept.
- Kety, S. S. (1957). Determinants of tissue oxygen tension. Federation Proceedings 16(2), 666-671. - The Krogh-Erlang steady radial diffusion-consumption solution and its pO2 profile.
- Popel, A. S. (1989). Theory of oxygen transport to tissue. Critical Reviews in Biomedical Engineering 17(3), 257-321. PMID 2673661. - Authoritative review of the Krogh model, its assumptions, and extensions.
- McGuire, B. J., & Secomb, T. W. (2001). A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand. Journal of Applied Physiology 91(5), 2255-2265. DOI 10.1152/jappl.2001.91.5.2255. - Michaelis-Menten O2 consumption M(P) = V_max P/(P_m + P) in the tissue cylinder.
- Goldman, D. (2008). Theoretical models of microvascular oxygen transport to tissue. Microcirculation 15(8), 795-811. DOI 10.1080/10739680801938289. - Modern review; parameter values for D, solubility, and consumption in muscle.
- Truskey, G. A., Yuan, F., & Katz, D. F. (2009). Transport Phenomena in Biological Systems, 2nd ed., Krogh tissue-cylinder example. Pearson. - Worked derivation of the Krogh-Erlang profile and the no-flux outer boundary condition.
- LeVeque, R. J. (2007). Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, sec 2.12 (ghost-node Neumann BC). DOI 10.1137/1.9780898717839.
- Press, W. H., et al. (2007). Numerical Recipes, 3rd ed., sec 2.4 - the Thomas algorithm for tridiagonal systems. CUP.
Solve the oxygen profile
Compute the pO2 field and lethal corner in your browser, with nothing uploaded to anyone's cloud. Save the run to Sandbox, attach it to a Worklog case, or share parameters through a Gate portal.