Biomedical Engineering Reference
In-Depth Information
In a numerical model the equivalent increment in the τ 2 time of
exposure is Arrhenius-based:
2.5.3 Example Multiple Damage
Numerical Model
Multiple damage processes may be studied in parallel to reveal
underlying trends and analyze planned heating protocols. Here
an example numerical model is used to illustrate the relative
transient development of several of the damage processes listed
in Table 2.1. A short CO 2 laser activation of 20 s followed by 10 s
of cooling was implemented in a 101 × 51 node axisymmetric
finite difference method (i.e., finite control volume) grid. The
model space included equilibrium boiling at 1 atmosphere pres-
sure—it also includes temperature- and water-dependent opti-
cal and thermal properties, but in this instance there was not
sufficient water vaporization to warrant recalculations. The CO 2
wavelength (10.6 μm) is dominantly absorbed in water with an
absorption coefficient of μ a = 792 (cm −1 ), and in the numerical
model it was assumed that 80% of the total absorption was in tis-
sue water (water = 50% concentration by mass) and 20% in resid-
ual (i.e., dry) tissue constituent proteins—as a result the effective
tissue absorption coefficient was 492 (cm −1 ). The 7 mm diameter
Gaussian profile beam had a total power of 0.5 W, resulting in a
center fluence rate of 2.6 (W cm −2 )—i.e., a maximum adiabatic
heating rate of 460 (°C s −1 ). The tissue was 2 mm thick with con-
vection and thermal radiation boundary conditions on both sur-
faces. The numerical model results are shown in Figure 2.9.
M T
τ= α+β+
[
P
]
(2.25)
e
2
where α = -152.35 = -ln{A}; β = 0.0109 (kPa −1 ); P = applied
stress (kPa); and M = 53,256 (K) = E a / R . The signs are
reversed in this expression because τ 2 appears in the denomi-
nator. Increments in t/τ 2 (i.e., dt 2 ) are calculated at the local
temperature as the model progresses. At the end of heating
the accumulated value for t 2 is converted to the shrinkage
in two normalizing steps:
e
a
(
ν−ν
ν−ν
)
m
f
()
ν= +
(2.26a)
a
(
)
1
e
m
where a = 2.48 ± 0.438, and ν m = ln{τ 1 2 } = -0.77 ± 0.26, and
then:
ξ= −ν +ν+ν +ν
(1
f
(
))[
a
a
]()[
f
bb
]
(2 .26b)
0
1
0
1
where a 0 = 1.80 ± 2.25; a 1 = 0.983 ± 0.937; b 0 = 42.4 ± 2.94; and
b 1 = 3.17 ± 0.47 (all in %). Values of ξ above about 60% indicate jelled
collagen, which in the case of tissue fusion is the desired result.
100
80
60
40
20
0
10
20
30
Time (s)
(a)
90
80
70
60
50
-3
-1
1
3
X (mm)
(b)
FIGURE 2.9 Laser beam heating axisymmetric FDM numerical model result. A CO 2 laser (λ = 10.6 μm) at total power of 0.5 W with Gaussian
beam profile, 2σ diameter = 7 mm for 20 s on tissue 2 mm thick with 50% mass fraction water. The effective laser absorption coefficient was
μ a = 492 (cm −1 ); and the beam center surface fluence rate 2.59 (W cm −2 ). (a) Beam center surface temperature history; 100°C reached at t = 18 s. (b)
Spatial distribution of temperature at the end of heating, 20 s. (c) Apoptosis/necrosis 10% and 90% damage contours. (d) Chinese hamster ovary
cell 10-90% contours. (e) BhK cell 10-90% contours. (f) Microvascular damage 10-90% contours. (g) Cardiac muscle whitening 10-90% contours.
(h) Skin burn coefficient (Diller (27) ) 10-90% contours. (i) Collagen birefringence loss 10-90% contours. (j) Collagen shrinkage 10-60% contours.
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