The Heart - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 113

When r 5 a,

1 065 J

:

σ θ 5 252 64 mmHg 438 9 mmHg

: 1 5 :

3

ð Þ

3 : 5cm

When r 5 b,

1 065 J

:

σ θ 5 252 64 mmHg 377 5 mmHg

: 1 5 :

3

ð 4cm Þ

The stress distribution within the ventricular wall is described by

1 : 065 J

r 3

σ θ 5 252 : 64 mmHg 1

with 3.5 cm # r # 4 cm.

This formulation predicts a differential work load carried out by the muscle cells along
the inner wall and the outer wall of the heart. In fact, it indicates that the cells along the
endocardium (inner wall) would produce more work than the cells along the epicardium
(outer wall). To meet this criterion, there should be an increased concentration of ATP, cal-
cium, and blood supply to the inner wall, which has never been measured in a laboratory
setting, although many have investigated this problem. In order to account for this, it has
been suggested that there is a residual stress within cardiac muscles, even under an
unloaded condition. These residual stresses and strains balance the uneven workload
requirement that has been previously summarized and therefore the need for increased
ATP, calcium, and blood. As this discussion and the example show, these formulas are
fairly limited in their accuracy due to the simplifying assumptions made about the
mechanical properties and the geometric properties of the left ventricle.
To couple the solid mechanics of heart motion and the fluid mechanics of blood flow
within the heart, the use of computation fluid dynamics would be required (see
Chapter 13). Fluid structure interaction and multiscale modeling should be incorporated
within this model to make the prediction as accurate as possible. There have been attempts
to compute an analytic solution to this problem, but they are generally overly simplified.
For instance, most will use an idealized geometry for the heart (similar to the spherical
shell described previously) combined with homogenous mechanical properties and incom-
pressible invisicid fluid flow. Without going into much detail about these models here, the
pressure and the fluid velocity in the ventricle can be solved by using Bernoulli's equation.
From Section 3.8, we learned that the Bernoulli equation can only be applied to very spe-
cialized flow situations and that the actual physiological scenario does not fit the assump-
tions necessary to apply the Bernoulli equation. Lastly, these solutions typically show that
the fluid is nonturbulent, which is not accurate from prior discussions on the generation
of turbulence. Most accelerating viscous flows through a constriction will become turbu-
lent at particular locations within the flow regime. These locations are typically character-
ized by flow separation and recirculation zones. A more thorough analysis of these types
of turbulent flows can be found in other classical fluid mechanics textbooks.