Biomedical Engineering Reference
In-Depth Information
match, and the skin burn coefficients from Weaver and Stoll pre-
dict substantially less thermal damage.
Also, the shape of the Breen et al. model prediction is notice-
ably different from the Arrhenius processes, as would be
expected from their respective functional forms. If the Breen
function is used to calculate the equivalent exposures for Ω = 1
at temperatures between 47°C and 60°C, the resulting plot on
Arrhenius axes is reasonably fit by a parabola—plainly not first-
order reaction behavior:
Briefly, Arrhenius measured the rate of hydrolysis of sucrose in
the presence of various acids in his original experiments. The
observations indicated that the temperature dependence of the
reaction rate was too great to be described by either the tem-
perature effect on the kinetic energy of the molecules or on the
dissociation of the acids. The measured reaction velocities at
temperatures
T
1
and
T
2
(K) were related by:
(
TT
−
)
BT T
TT
(
−
)
=
B
TT
21
k
k
|
|
21
21
T
(2 . 21)
=
e
e
2
21
2
1
−
1
+
T
1
8
5
3
2
ln{} 1.562
τ= ×
10
9.384
×
10
1.412
×
10 (
r
=
.987).
T
T
where Arrhenius's original notation has been revised to that used
in this chapter for clarity, and
B
is an experimentally determined
constant. The reason for the particular regrouping in the right-
hand expression will become clear in the following discussion.
R
CEM
in Equation 2.20 is the ratio of exposure times required
to result in the same survival for a 1°C rise in temperature
(note that τ
2
/τ
1
=
k
1
/
k
2
, an inverse relationship). That is, for
T
2
=
T
1
+ 1 (K) and Ω
2
= Ω
1
, then:
(30)
(2.19)
The Arrhenius model is based on well-established fundamental
physical principles, so it is difficult to see any real advantage to
the reformulation in the style advocated by Breen et
al.
2.4 Comparative Measures for thermal
Histories: thermal Dose Concept
−
E
RT
a
−
E
B
TT
−
=
τ
τ
Ae
a
1
R
2
1
=
=
e
RT
(
T
+
)
=
e
.
(2.22)
11
10
For approximately the past three decades the standard method
for assessing hyperthermia treatment effectiveness has been
the thermal dose unit. The concept of thermal dose units, in
the form of cumulative equivalent minutes (CEM) of exposure
at 43°C, dates at least from the pioneering work of Sapareto
et al. in 1978
(32)
and is also very well described in their chap-
ter in the subsequent 1982 topic.
(31)
Thermal dose units derive
from the observation that above 43°C for many cell lines stud-
ied a similar level of damage resulted in approximately half the
time when the temperature increased by 1°C.
(31,32,56)
A CEM
value in excess of an accepted threshold is considered indica-
tive of a successful treatment. The applied thermal dose, CEM,
is calculated from:
CEM
−
E
RT
a
Ae
2
The relationship to Arrhenius's original work is now appar-
ent, and the value of
B
is seen to be
E
a
/
R
. At 44°C
R
CEM
for the
Chinese hamster ovary cells is 0.479 (see Table 2.3), very close to
0.5. For uniform computation of CEM values the representative
value for
R
CEM
has historically been taken to be 0.5 above 43°C
and 0.25 below 43°C to match the data for the several cell types
originally listed.
(31)
2.4.2 Example process Calculations
Values for
R
CEM
are given in Table 2.3, computed at
T
1
= 317.16 K
= 44°C for the thermal damage processes previously discussed.
Note that no information about the process temperature offset,
A
, is contained in
R
CEM
; consequently, by default all damage pro-
cesses are referred to the 43°C reference point in this measure.
R
CEM
depends more strongly on the activation energy,
E
a
, than it does on
the temperature. At 43°C
R
CEM
= ex p(1.199 × 10
−6
E
a
), while at 50°C
R
CEM
= ex p(1.14 8 × 10
−6
E
a
), only very slightly different.
τ
N
∑
∫
(2.20)
(43)
−
T
(43)
−
T
CEM[
=
R
]
t
=
[
R
]
dt
i
43
CEM
i
CEM
i
=
1
0
where the exposure is either in discrete intervals,
t
i
, or continu-
ous, and
R
CEM
= 0.5 is typically used above 43°C (and
R
CEM
= 0.25
below 43°C). The authors suggest the use of a normalized time
of exposure in order to provide a means for quantizing hyper-
thermia treatments, CEM, or cumulative equivalent minutes at
43°C. In that way, an arbitrary transient temperature history
can be directly compared to a constant temperature exposure
at 43°C.
2.5 applications in thermal Models
Implementation of Equation 2.6 (the Arrhenius model) or
Equation 2.20 (the CEM calculation) in a thermal model is
fairly straightforward, but may require some care. For exam-
ple, the value of Ω increases very rapidly above threshold
temperatures estimated from Equation 2.8, and may quickly
saturate the computer arithmetic if not limited. Values
of Ω above about 10—i.e.,
C
(τ) ≤ 4.5 × 10
−6
—yield no new
information since the damage process is plainly saturated;
2.4.1 Foundation of thermal Dose Concept
Thermal dose units may be seen to derive directly from
Arrhenius's original description, as described in the summary
of his pioneering work in the first chapter of Johnson et al.
(24)
