The Krogh Model
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Possibly the first attempt to describe tissue
oxygenation was by
Krogh (1). He considered a tissue circle
with a central capillary and asked his mathematician Erlang to solve for the
diffusion equation:
| P = P0 + |
M |
{
r2 − 2R2ln(r)} |
| 4K |
better written as:
| P = Pc + |
M |
{
r2 − rc2 − R2
ln( |
r2 |
)} |
| 4℘ |
rc2 |
where Pc is capillary rim O2 pressure, M is tissue oxygen consumption,
℘ is oxygen permeability of the tissue - identical to the K proposed by Krogh,
r distance from the center,
where the capillary is, with radius rc, and R radius of the circular region,
area A=πR2, and the P0 is directly related to Pc.
Because of the circular boundary, there is radial symmetry which makes P only dependent
of the radial distance r, and the flux towards the boundary is:
| J = − ℘ |
∂P |
= ½M { |
R2 |
− r} |
| ∂r |
r |
which clearly is zero at the boundary r=R so that all oxygen is consumed in the
circular area.
Later, Kety (2) took the step to
extend the cylinder along the capillary, and this became known as the
Krogh cylinder. That is possible because in each plane an oxygen amount of
MπR2 is consumed per time so removed from the capillary:
| MπR2 =
− πrc2 cHb |
dS |
= − πrc2 cHb v |
dS |
| dt | dz |
where cHb is oxygen binding capacity of the blood, S is saturation
and v blood velocity. This means that S decreases linearly from arterial (a)
side (z=0) to end-capillary (e) side:
Some remarks have to be added.
- In most textbooks, the Krogh cylinder is not properly scaled. The figure here more
reflects an actual situation.
- The main advantage of the Krogh equation is, that an estimate can be made of
the minimum capillary pressure to supply the full circular area. If it can no
longer been maintained, near r=R a so-called Dead Zone will emerge.
- The Kety approach allows to estimate if blood supply is sufficient.
- All these approaches ignore, that capillary rim PO2 is lower than
blood PO2. The difference is called Extraction Pressure and is
a fixed value for a given situation so can easily be accounted for.
(1) Krogh A: The number and distribution of capillaries
in muscle with calculations of the oxygen pressure head necessary for supplying the
tissue. J Physiol (London) 52: 409-415 (1919).
(2) Kety SS: Determinants of tissue oxygen tension.
Fed Proc 16: 666-670 (1957).
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