Exchange Coefficient

Examining the velocity and oxygen concentration contour results for the terminal acinar cluster, in Sections 4.4.2 and 4.4.3, reveals each alveolus in the cluster experiences the same conditions. Therefore, each alveolus can be treated equally, without any special averaging required. With the transport of oxygen from the inlet to the wall in the pore level simulations elucidated, the average exchange coefficient for the oxygen diffusion within the lungs can be determined (average in the sense it is derived from results of the entire terminal cluster). The exchange coefficient, k in m/s, is defined as

! =! !!

!!_! !_!,! −!_!,! (5.11)

where !! is the solubility of species A in tissue in kg/m3, A is the average surface area

available for exchange during respiration, !_!,! and !_!,! are the concentrations of species A in mol/kg at the inlet and wall, respectively, and !_! is the exchange rate of species A in mol/s. The exchange rate for oxygen in one breath can be calculated by

!_! =!!!

! =!!! (5.12)

where t is the length of time exchange occurs per breath in seconds, f is the frequency of respiration and !_! can be found in general as:

!_! =! !_!∆! !_!,!"!!"#$ −!_!,!"!!"#$,! !"

!!!

(5.13)

where i is the number of time steps over the exhale, !_! is the mass flow rate at each time step in kg/s, ∆! is the length of each time step and !!,!"!!"#$,! is the mass flow averaged

concentration at the outlet in mol/kg. As we are seeking a constant value of k suitable for all calculations, it is useful to consider the maximum amount of oxygen that could be absorbed per breath as supplied by the air:

where ∆!_! is the tidal volume of the terminal alveolated cluster studied in Chapter 4, ! is the density of the air mixture. In this manner, Eq. 5.11 combined with Eqs. 5.12 and 5.14 becomes:

! =!!"∆!!

!!_! (5.15)

where ∆!!and A are both parameters from the terminal alveolated cluster, and the

frequency f is the only parameter left to be determined. This coefficient can be recast into a resistance as:

! = 1 !!_!" =! !_!! !!_!"∆!_!∙ 1 ! (5.16)

where the resistance to diffusion is seen to be inversely proportional to the frequency of respiration. This suggests when the rate of respiration decreases, the resistance to diffusion increases, i.e. one breath every ten seconds will absorb less oxygen per breath than one breath per second, which is not reasonable.

It is more reasonable to expect the exchange coefficient k is constant with respect to respiration frequency, and more dependent on tidal volume. Then, since the model to this point is based on the maximum possible exchange for a given ∆ !_! and ∆!_!, we propose this frequency should correspond to the maximum frequency where complete exchange can occur.

The time scale for the oxygen diffusion through the blood-gas barrier is

!~!!

! (5.17)

where L is the 2 !m thickness of the barrier and D is the diffusion coefficient of oxygen through tissue at 2.5×10-5 cm2/s [34]. The time scale for diffusion is approximately 0.001667s. Since this is longer than the time scale of diffusion through the alveolar air, the oxygen diffusion should be dependent on the properties of the tissue. Therefore, the

frequency at which the maximum exchange per breath can occur within the lungs for a given tidal volume should be determined based on the properties of the lung tissue. Rapid breathing, or tachypnea, is defined as a respiration rate greater than 20 breaths per minute, or one breath every three seconds (0.33 Hz). This frequency can also be determined by calculating the time it takes to absorb the full amount of oxygen available per breath. This can be done by multiplying the flux of oxygen into the tissue by the surface area available for exchange, by the solubility of oxygen in the tissue over the thickness of the blood-gas barrier. The maximum amount of oxygen in moles, available for exchange can then be divided by this value. Therefore the minimum time it takes for the full amount of oxygen available per breath to be absorbed by the lungs is

! =_!!_!!"#! !!∆!!

(5.18) where !_!"# is the maximum moles available per breath (4.4868×10-10 mol from the pore level simulation with a 15% tidal volume), A is the average area available for exchange in the simulated acinar cluster from Chapter 4 (1.662×10-5 m2), D is the diffusivity of oxygen in tissue at 2.4×10-9 m2/s [34], !_! is the solubility of oxygen in tissue at 0.004406 kg/m3 [35], L is the thickness of the blood-gas barrier at 2 !m, and the change in concentration is the same used in Chapter 4 with the units of mol/kg. Therefore the minimum time to absorb the full amount of oxygen available per breath for a 15% tidal volume is 2.91 s. This makes the maximum frequency at which full absorption is

possible, !_!, for a 15% tidal volume approximately 0.34 Hz, the same value mentioned

before as rapid breathing.

This maximum frequency can be integrated into Eq. 5.15 giving

!=!!!!∆!!

!!_! (5.19)

This gives the resistance to diffusion in terms of only one changing variable, ∆!_! or tidal volume. As seen in the following equation, the new k term changes Eq. 5.16 to

! = !!! !!_!"∙

1