The Multi-Krogh Model

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In the Krogh model, a single capillary supplied a cylinder, a circular area in each cross section:

P = Pc + M { r2 − rc2 − R2 ln( r2 )}
4℘ rc2
but for an extension to tissue this equation first has to be analyzed along Fick's second law ∂c/∂t = ℘∇2P. For the r2 term, this yields M so that this term is compensating tissue oxygen consumption, in an area A=πR2. Consequently, the ln-term represents the other feature, the capillary. So, for a tissue cross section with N capillaries, diffusion can be described by:
P = P0 + M { Field(r) − Σ N  Ai ln( |r − ri|2 )}
4℘ i=1 π rci2
where the ith capillary is located at ri and has radius rci supplying an area Ai. Note the underlining indicating 3-dimensional entities. The Field() term has to compensate consumption:
M = ℘∇2P = ¼M ∇2 Field(r) => ∇2 Field(r) = 4
which is |r|2 for a circular cross-section but can be found for other shapes too – see this document. This may be a good model if in the consecutive layers the ri are almost the same, so parallel capillaries, as in muscle tissue. Then, the problem is, how to find the constants P0 and Ai. These are N+1 unknowns, so we need N+1 boundary conditions. N of these are, that P must equal (the averaged) capillary rim pressure Pck at the kth capillary rim:
Pck = P0 + M { Field(rk) − Σ N  Ai ln( |rk − ri|2 )}
4℘ i=1,i≠k  π rci2
and the last one, that supply must match consumption:
A = Σ NAi
i=1
where A is the area of the entire cross section.

Mostly, the capillary rim pressures are only known in a specific cross section, practically, the starting one. Since for each capillary the supply area Ai is known, the drop in Sci so Pci can be calculated similar as for the Krogh model:

M Ai = − πrci2 cHb,i dSi = − πrci2 cHb,ivi dSi
dtdz
and from these the next level P0, Ai, and so on. Note, that Si is no longer linear since the Ai will change.

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