Flow in the Lungs - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 260

that the stress-strain relationship was actually independent of the loading frequency (e.g.,
breathing rate) and therefore was pseudo-elastic. In these same experiments, the results
were fit to a two-dimensional strain energy function fairly accurately. From this experi-
mental work, the circumferential stress (S xx ) and the longitudinal stress (S yy ) can be
defined as

S xx 5 a 1 E xx 1 0 : 6429E yy

ð 9 : 2 Þ

S yy 5 a 2 E yy 1 0 : 6429E xx

where E xx and E yy are the Green's strains in the circumferential and longitudinal direction,
respectively, and a 1 and a 2 are material properties. Other experimental work has shown
that the relationship between pressure (blood pressure minus the pleural pressure) and
vessel diameter is linear within the pulmonary blood vessels. In comparison, for systemic
vessels this relationship is highly non-linear. More than likely this is due to the fused basal
laminae layer which intimately links the pliant blood vessels to the stiffer lung tissue.
Therefore, linear elastic relationships can be used when determining the blood vessel
cross-sectional area change as a function of hydrostatic pressure, interpulmonary pressure,
and intrapleural pressure.
Changes in the interpulmonary pressure and the blood hydrostatic pressure can also
affect the size of the alveoli as well as the distance that separates the gas in the alveoli and
the gas in the blood capillary. Under normal conditions, the alveolar size remains constant
as a function of blood hydrostatic pressure but varies with interpulmonary pressure. In
simple terms, the surface area of the alveoli will follow the inverse of Boyle's Law, so that
when the interpulmonary pressure increases, so does the size of the alveoli. This relation-
ship is a direct relationship so that mass is conserved during lung expansion. The separa-
tion distance is a slightly more difficult problem, because it will be affected by the
interpulmonary pressure as well as the capillary hydrostatic pressure. Let us define

Δ P 5 P capillary 2 hydrostatic 2 P interpulmonary ð 9 : 3 Þ

Using Equation 9.3 , we can define the separation distance between the alveolar gas and
the capillary gas (this is a measure of the respiratory boundary thickness and not the exact
diffusion distance). The values for separation distance have been quantified experimen-
tally and agree with the following discussion. If Δ P is negative and less than negative

0.7 mmHg, the separation distance is equal to 0. When Δ P is between negative 0.7 mmHg

and 0 mmHg, the separation distance increases from zero to its nominal resting value of
approximately 0.5

Δ P increases from zero, the distance increases linearly as well to
some limiting value of approximately 1.3

μ m. As

m at a pressure of approximately 30 mmHg.
Mathematically, this can be represented as a piecewise continuous function:

μ

8
<

0 Δ P ,2 0 : 7 mmHg

0 : 5 μ m

0 : 5 μ m 1

7 mmHg Δ P

2 0 : 7 mmHg , Δ P , 0 mmHg

0 :

h ðΔ P Þ 5 ð 9 : 4 Þ

m
75 mmHg Δ P

2 μ

:

: μ 1 , Δ P ,

0 5 m 0 mmHg 30 mmHg

1 : 3 μ m Δ P . 30 mmHg