Biomedical Engineering Reference
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where A is the cross-sectional area, (A
0
the relaxed area), T the longitudinal wall
tension and c a damping factor. U represents the circumferential wall stress, which
was expressed using the thick wall model [
3
] often used in arterial mechanics:
ð
a
ext
a
int
Þ
U
ð
Da
ext
Þ¼
EDa
ext
2
ð
1
r
2
Þ
a
int
a
ext
;
ð
52
Þ
where a
ext
and a
int
are the external and internal radii, Da
ext
the change in external
radius caused by the pressure difference U, and r is Poisson's ratio. E is the
Young's modulus for the wall material. To simulate the effect of pumping, either
the natural (resting) radius or E can be changed; both approaches were investi-
gated. The computational work was supported by experimentation to determine
appropriate values for coefficients such as T and c. Experimentation was also used
to determine the pumping model for E,
E
ð
t
Þ¼
E
relaxed
þð
E
contracted
E
relaxed
Þ
n
ð
t
Þ;
ð
53
Þ
and a similar model for the zero-pressure radius a
0
; for the pumping function
n
ð
t
Þ2½
0
;
1
, two formulations were used, a simple sine wave with period t
p
, and a
shorter sine wave with pause at relaxation (total time period t
p
). Phase differences
between the pumping function at adjacent nodes enabled the simulation of
orthograde and retrograde contractile waves. Results for these simple pumping
functions were entirely consistent with experimentally observed results, generating
a sawtooth output which matched observed radius variations. Pumping worked
best with contraction being nearly simultaneous in all nodes within a lymphangion,
and differences between orthograde and retrograde contractile waves were minor.
For sufficiently high reverse gradients (i.e. edema), pumping merely served to
reduce the flow rate, in agreement with the computational results of [
56
] reported
earlier. Numerically the explicit algorithm placed a stringent constraint on the
computational timestep which could be used, governed by the propagation of
pressure/contractile waves in the system; the additional terms T and c also acted to
smooth the solution somewhat, producing less erratic results.
4.3 Higher Dimensions
Full solution of the Navier Stokes equations in 2D or 3D, except in very limited
cases, is only really possible numerically, a fact which has led to the development
of the subject of Computational Fluid Dynamics or CFD. CFD has had a major
impact on biomechanics, in both cardovascular and respiratory [
8
] areas. Com-
monly-used is the Finite Volume method (FVM), in which the domain of interest
is divided into a multitude of small cells or control volumes; the assembly of these
is referred to as a mesh. The Navier-Stokes equations are integrated over each cell,
and this, combined with application of Gauss' theorem to convert the transport
term in the Navier-Stokes equation into fluxes across the cell faces, converts these
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