Biomedical Engineering Reference
InDepth Information
The classical method to determine damage process coefficients
is to expose the tissue to constant temperatures for measured
times and determine Ω by some quantitative measure, such as cell
survival fraction or vital stain/fluorescence intensity. Looking at
Equation 2.1 and rearranging, the logarithm of the result follows:
2.3.3 Example thermal Damage processes
Thermal damage has been measured in a very large number of
processes over the past four decades. A few representative pro
cesses have been selected for illustrative purposes in this section.
More complete reviews of this literature may be found in sev
eral recent publications.
(1,6,29,30)
The process parameters for the
examples as described may be found in Table 2.1.
τ− Ω=
1
E
R
−
a
ln{} ln{}
ln{} ln{}
A
=
τ
(2.12)
eq
T
2.3.3.1 Cell Survival Studies
In the classical experiments of the thermal sensitivity of Chinese
hamster ovary cells by Sapareto et al.
(31,32)
“survival” is indicated
by the ability of the cells to continue to form colonies. The sur
viving fraction at constant temperature was measured
vs
time
analogously to Equation 2.6. The slope of the survival curve,
D
0
corresponds to 1/
k
:
where τ
eq
= τ/Ω, the time of an equivalent exposure at
T
for which
Ω = 1. An ensemble of such experiments is plotted on Arrhenius
axes—ln{τ
eq
} on the ordinate and (1/
T
) on the abscissa—for
which a leastsquares fit has slope =
E
a
/
R
and intercept = ln{
A
}.
The determination is very sensitive to small uncertainties in
temperature owing to the hyperbolic dependence in the expo
nent; consequently, extremely careful attention must be given
to accurate temperature measurements, above all other consid
erations. In all of these experiments, the linear regression line
effectively holds the short end of a very long “stick,” as it were,
and the uncertainty in the intercept, A, is huge as a result.
tt
DT
−
S
N
0
C
C
()
(0)
τ
−
=
e
()
=
e
−−
kt t
(
)
=
(2.15)
0
0
0
t
where
N
0
cells are counted at time
t
0
and
S
of them survive at
time
t
. The Arrhenius plot of
D
0
vs
T
has an obvious break point
at
T
= 43°C in its data, and this temperature is used as the ref
erence temperature in hyperthermia work. A break point in an
Arrhenius plot indicates that different thermal processes are
dominant above and below the break point temperature.
For the Chinese hamster ovary cells above 43°C, the
Arrhenius coefficients work out to:
A
= 2.84 × 10
99
(s
−1
) and
E
a
= 6.18 × 10
5
(J mole
−1
), with a critical temperature of 51.4°C.
Figure 2.3 plots the predicted surviving fraction
vs
time at 44°C
for these cells; the response is characteristically sigmoidal, as has
been described. The curve shifts to the left as the temperature
increases.
In cell survival studies involving longer term heating at lower
temperatures it is often more instructive to use the time at 43°C
to reach Ω = 1 as a comparative parameter, rather than
T
Tcrit:
:
2.3.2.3 transient Experiments
The case of transient experiments in which only a single assay
can be determined at the conclusion of the experiment is an espe
cially thorny one. Unfolding Equation 2.1 to a suitable form for
calculation from the experiment results is intractable. There is an
approach that can be used in this case, however.
(1, 3)
Each experi
ment has a measured transient thermal history,
T
i
(
t)
, and corre
sponding level of damage, Ω
i
, for which
E
a
and
A
are unknown.
For each experiment in an ensemble, a selected segment of the
ln{
A
} 
E
a
plane can be scanned for values that yield the measured
Ω
i
from the transient history,
T
i
(
t
). Define a “cost” function that is
minimized when A and
E
a
give the correct integral, such as:
E
RT t
−
a
τ
∫
()
Ae
dt
(2.13)
i
Cost
=
ln
0
.
{
}
E
Ω
a
τ=
exp
−
ln{}
A
(2.16)
i
43
3
2.629
×
10
The locus of points in the ln{
A
} 
E
a
plane that yield the
required Ω
i
value lies along a straight line. Rearranging Equation
2.12 one obtains:
1.25
ln{}
1
τ=
E
R
−
a
ln{ .
A
(2.14)
1.00
eq
T
eq
0.75
C
(
τ
)
The slope of the solution line for the
A

E
a
pairs gives
T
eq
, and the
intercept τ
eq
; the temperature and exposure time, respectively, for
an equivalent constant temperature exposure for which Ω = 1. It
is not difficult to estimate the likely range of ln{
A
} values that will
match a selected search range for
E
a
from Equations 2.11a,b above.
The ensemble of equivalent constant temperature points for the
experiment series is then plotted on standard Arrhenius axes to
determine
A
and
E
a
for the process under study.
0.50
0.25
0.00
1
10
100
1,000
10,000
Time (s)
FIGURE 2.3
Cell survival fraction for asynchronous Chinese ham
ster ovary cells
(31,32)
at 44°C.