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dr

Km

+C Dg

which upon integration gives

(3.27)

(3 .28)

Since symmetry exists about the centre of the follicle

(dC/dr=O,

at

r=O),

and therefore

K, =0. Rearrangement of equation

3.28

gives,

(3 .29)

which upon integration gives,

(3.30)

Since at

r=rj, C=Co,

then

Substitution of 3.3 1 into 3.30 and rearrangement gives, C Rgmax 2 2 K _m In-_C + (C-C ₀ ) = --- (_6D r - r _f ) o g (3 .3 1 ) (3.32)

Equation 3 .3 2 gIves the oxygen concentration at any position r in the pre-antral follicle when Michaelis-Menton oxygen consumption kinetics are considered. Equation 3 . 3 2 cannot be solved explicitly, and hence requires an iterative solution (for the purposes of comparison, equation 3.32 assumes no voidage). Setting equation 3 .3 2 equal to zero and solving for the distance from the follicle centre at which the fol licle becomes anoxic is not possible in this case. However, equation 3.32 can be used to provide a comparison with the solution of equation 3 . 1 4 for the oxygen profiles through the pre-antral follicle.

Figure 3 . 1 6 compares the solutions of equation 3 . 1 4 to that of equation 3 .3 2 at both high and low values of Km for a large pre-antral follicle. All parameter values other than rf are at their nominal values (see Table 3 .3).

Figure 3 . 1 6 shows that when Michaelis-Menton kinetics are considered oxygen concentration reduces more gradually at low oxygen concentrations than when the

consumption rate is constant. At a high Km value this is more dramatic.

Using Michaelis-Menton kinetics the oxygen concentration tends toward zero but does not reach a true anoxic state (although for biological purposes it is still essentially anoxic). This suggests that even under hypoxic conditions some oxygen may still be able to reach the oocyte surface. However, such small oxygen concentrations are unlikely to be of any real consequence for the oocyte. Using a high

Km value oxygen only penetrates the follicle an extra

5

)..tm (after which its concentration is < 1 x 1 0-5mmol.m-3, which can be considered practically zero) when

compared with constant respiration. Thus, even with a favourable Km value the effect of Michaelis-Menton kinetics is small. Without any real knowledge of the oocyte's respiratory activity in a pre-antral follicle it is difficult to speculate any further than this.

Although Michaelis-Menton kinetics allow oxygen to penetrate further into the follicle, what may be of more significance is the point at which Rgrnax can no longer be sustained. This concept is shown in Figure

3 . 1 7.

Figure

3 . 1 7

illustrates that there exists a critical oxygen concentration below which Rg cannot be sustained at Rgrnax. If

Ccril is defined as the oxygen concentration at which Rg = 0. 99 Rgrnax then equation

3 .26

can be solved (iteratively) over the feasible range of

Km (0.0005

to

0 .003

mol .m­

\

to give a feasible range for Cril. This gives a range of CCril from

50

�M to

297

�M. This range is very wide, particularly when compared with the arterial blood oxygen concentrations

( 1 30

�M to

1 50

�M). The upper range of CCril would suggest that even cells experiencing the high concentrations of alierial blood will not be able to respire maximally. With such a wide range in

Km

little can be gained from pursuing a model which includes Michaelis-Menton kinetics. Still, equation

3 . 1 4

can be set equal to

CCril and solved for r giving equation

3 .3 3 .

r =

(3.33)

where Dg = Deff

Or, if voidage is to be included equation

3 .33

can be solved in the same way giving

equation

3 .34

r =

_{(3 .34)}

where Dg = Deff

1 2 0 - '? E 0 _{9 0} E c 0 +' � ... c Cl) (.) 60 c _{high Km} 0 (.) c Cl) C'l >- >< 0 _{low Km} 3 0 constant Rg o 1 7 5 1 8 5 1 9 5

Dista n ce fro m fo l l icle ce ntre (Jlm)

Figure 3 . 1 6 Oxygen concentration profiles i n a large pre-antral follicle under the

assumption of constant granulosa cell oxygen consumption (Rg) compared with Michaelis-Menton kinetics.

2 � c o :p a. E :::I tJ) C o u c Cl) C'I

�

o Figure

3 . 1 7

O. 99Rgmax Rgmax o o Oxygen concentration

Concentration dependence of oxygen consumption described by

Michaelis-Menton kinetics (not to scale).

Equations 3 .33 and 3 .34 can be used to calculate the distance from the centre of the follicle at which maximal respiration is no longer possible under the assumptions of no voidage and voidage respectively. Alternatively Ccril (and hence

Rg)

could j ust as easily be any concentration which may have some significance. For example, it may be the concentration at which cells are known to be compromised, and this could possibly be somewhat lower than Ccrit for marginal sub-maximal respiration.

In spite of the lack of information about CcrU some useful general statements can sti l l be made. If voidage i s considered, there will exist a critical follicle size beyond which maximal respiration can never be achieved for any given voidage level. Similarly, at any given fol licle size a certain critical voidage will be required if respiration is to be maximal.

3.3.6 The assumption of complete vascu larisation

The assumption that each follicle is surrounded completely by its own vascularisation is not always true for antral follicles, and is unlikely to always be the case for pre­ antral fol licles. Both the amount and distribution of vascularisation around a fol licle

are likely to be important.

If a follicle is considered to be surrounded by areas of vascularisation and areas of no

vascularisation which are distributed symmetrically, with each patch being infinitely

small, then reducing the value of Co down from its arterial value can be used to

simulate a reduced area of vascularisation.

This concept is described by equation 3 . 3 5 .

%vasc

C

(1 00 -%vasc)

C

Omean =

C

arterial X 1 00 + novasc. X

1 00 (3 .35)

Assuming in areas of no vascularisation the oxygen concentration is zero, then Cnovasc =

O.

Hence, equation 3.35 can be solved to give the % vascularisation.

lOOC %vasc =

Omean

In document "Monumentos con pies" Arte público conmemorativo y escultórico en el centro de Bogotá (página 88-96)