Consider the Krogh cylinder model for oxygen transport in skeletal muscles described in Example 1.28.
(a) At what radial position in the tissue surrounding the capillary entrance is the oxygen partial pressure down to 60.0 torr?
(b) At a position along the capillary where the oxygen partial pressure in the blood is 25 torr, the lowest oxygen partial pressure in the surrounding tissue is 10.0 torr. Estimate the corresponding value of the oxygen demand, \(M_{0}\).
(c) For the conditions described in part (b), estimate the oxygen flux leaving the capillary wall.

Data From Example 1.28:-

In 1922, August Krogh proposed that oxygen was transported from capillaries to surrounding tissue by passive diffusion. He formulated a simple geometric model to describe this phenomenon. The Krogh model is a simplification of a capillary bed; a group of parallel identical units, each containing a central capillary and a surrounding cylinder of tissue, as shown in Figure 1.12. The goal of the Krogh model is to predict the concentration of oxygen as a function of position within the tissue cylinder. Each tissue cylinder is assumed to be supplied with oxygen exclusively by the capillary within it. Oxygen diffusion in the axial direction is neglected, as evidenced by the fact that the oxygen concentration gradient is much steeper in the radial direction than in the axial direction.
Applying Fick’s law of diffusion in the radial direction (Rc ≤ r ≤ Rt) and conservation of mass lead to the following equation for oxygen diffusion in muscle tissue (McGuire and Secomb, 2001):

where ca is the oxygen concentration in tissue. In studies of oxygen transport, it is customary to express the concentration in tissue in terms of an equivalent oxygen partial pressure, pa = HcA, where H is a modified form of Henry’s law constant. Then, equation (1-125) can be written in terms of the oxygen partial pressure in tissue as 

where K (known as the Krogh diffusion coefficient) = D_AB,eff/H, and M(p_A) is the
oxygen consumption rate per unit volume of the tissue cylinder. Oxygen consumption
in skeletal muscle is usually assumed to follow Michaelis-Menten kinetics, a
model that describes the kinetics of many enzymatic reactions, given by 

where M₀ is the oxygen demand (i.e., the consumption when the oxygen supply is not limiting) and p₀ is the partial pressure of oxygen when consumption is half of the demand. The Michaelis-Menten equation reduces to first-order kinetics for values of p_A much smaller than p₀, and to zero-order kinetics for values of p_A much larger than p₀. For this example, we will assume zero-order kinetics; therefore, M(p_A) = M₀. The two boundary conditions needed to solve equation (1-126) are 

1. Neglecting the oxygen-diffusion resistance of the capillary wall, at the capillary–tissue interface, the oxygen partial pressure in the tissue is approximately equal to the average partial pressure of oxygen within the blood, P_{A b}; that is, PA^(R _C ⁾ = P_A,b.

2. It is assumed that no oxygen is exchanged across the outer boundary of
the tissue cylinder, so that

The complete solution for zero-order kinetics is

Equation (1-128) can be used to predict whether, under a given set of parameters,
there are hypoxic regions—where the partial pressure of oxygen is less than 1 torr—
in the tissue cylinder. Table 1.3 summarizes parameter values used in the Krogh
model.
To illustrate the use of equation (1-128), we use the parameters of Table 1.3,
with an oxygen consumption of 0.4 cm³ O2/cm³-min, and estimate that the lowest
oxygen partial pressure in the tissue surrounding the capillary entrance is 49.9 torr
at r = R_t. Therefore, all the tissue at this position is exposed to a healthy oxygen
concentration with no hypoxic regions. Moreover, since p_A is everywhere much
higher than p₀, the assumption of zero-order kinetics is validated.

However, as blood flows along the capillary, oxygen is extracted by the tissue and the oxygen partial pressure in the blood declines rapidly. The oxygen partial pressure in the tissue also declines rapidly and a point is reached where zero-order kinetics no longer applies. The complete form of the Michaelis-Menten equation must be used, requiring a numerical solution of equation (1-126). McGuire and Secomb (2001) showed numerically that, under the set of conditions considered in this example, as much as 37% of the tissue was hypoxic.

Transcribed Image Text:

1 d DABEff T dr de A d|| r dr =RA (CA) (1-125)