Biomedical Engineering Reference
In-Depth Information
so that
C*
E
RT ()
∆H*
a
τ
C
C
(0)
()
Enthalpy
(2.6)
ln
=
Ae
dt
=Ωτ
()
C
τ
0
∆H
and the physical significance of the damage parameter, Ω, as
used in Equation 2.1 is now easily seen. Equation 2.6 constitutes
the theory of absolute reaction rates. Using this approach, the
remaining undamaged tissue constituent, surviving cell frac-
tion, or probability of thermal damage is
Native
Denatured
State
FIGURE 2.1 Activation energy barrier, Δ H *, between native and
denatured states. Activated complex can either relax back to the native
state or progress to denatured at the net forward velocity, k .
−Ω
−Ω
C
() 100
τ=
e
(%)or, theprobability is
P
(%)
=
100[1
e
].
activated complex may either relax to the inactivated native state
or may proceed to a denatured state, as depicted in Figure 2.1, at
overall denaturation forward velocity, k .
In the figure the enthalpy represents the total energy; the
enthalpy of activation, Δ H *, should not be confused with the
sensible reaction heat, Δ H . The reaction velocity, k , is in turn
given by the Gibb's free energy of activation, Δ G *:
(2.7)
2.3.2 arrhenius process Functional
Behavior and Determination
of process Coefficients
The functional behavior of Equations 2.1 and 2.6 warrants some
discussion at this point. Additionally, the standard methods to
determine damage process coefficients are worthy of review.
There are three possibilities: (1) the damage process can be mea-
sured in real time as it develops, such as fluorescence intensity,
in which case determining the appropriate coefficients, A and E a ,
amounts to a curve-fitting exercise; (2) the result can only be
determined a posteriori , as by histologic assay, but the experi-
ments are at constant temperature, or nearly so; or (3) the result
can only be determined a posteriori , but the experiments are
plainly transient, for example, as in a laser heating experiment.
This section treats these issues in turn.
*
G
RT
.
(2.3)
ke
=
The activation enthalpy, Δ H *, includes both the Gibb's free
energy of activation and the entropy of activation, Δ S *:
*
* *
=+
HGTS .
(2.4)
Consequently, in terms of the activation entropy and enthalpy,
the Eyring-Polyani equation gives:
*
*
*
S
R
H
RT
S
R
E
RT
=
+
1
RT
Nh
RT
Nh
a
(2.5)
k
e
e
e
e
P
P
2.3.2.1 Functional Behavior
The damage parameter, Ω, makes a convenient measure of the
process progress; Ω = 1 is a convenient calculational reference
point and is often referred to as the “threshold.” However, it
should be borne in mind that by the time Ω = 1 the process is
63.2% complete—hardly the “threshold” for the process. Using
Ω = 1 as the reference point, the “threshold temperature” in a
constant temperature exposure, T TH (K), is given by:
where N = Avogadro's number (6.023 × 10 23 mole −1 ), and h P =
Planck's constant (6.625 × 10 −34 J s mole −1 ). Here the activation
enthalpy has been truncated to the activation energy, E a , as an
approximation—in fact, Δ H * = E a - iRT , where i is the order of the
reaction. In practice iRT is small compared to E a for a first-order
damage process—i.e., the 1 in Equation 2.5 is small compared to
Δ S */ R . Note also that the fraction preceding the entropy expo-
nential varies little over the few K temperature range that typi-
fies the active region of a damage thermal history. Consequently,
the entire preexponential factor is usually treated as a constant,
A , as in Equation 2.1. The physical significance of A is that it is an
indication of the collision frequency between reactants, or some
analogous quantity in a unimolecular reaction.
The solution of the governing differential equation
(Equation 2.2) is determined by an appropriate integrating fac-
tor, and given by
E
RA
a
T
=
} n{ }] .
(2.8)
TH
[ln{
For a different reference damage level, Ω ref , the threshold con-
stant temperature is:
E
a
T
=
}] .
(2.9)
TH
RA
[ln{
} n{ } n{
+τ−Ω
ref
{
}
Finally, an effective characterizing parameter for a higher tem-
perature damage process is its “critical temperature,” T Tcrit (K),
τ
kdt
CCe
()
τ=
(0)
0
Search WWH ::




Custom Search