Biomedical Engineering Reference
In-Depth Information
receptors and their interactions need not be modelled and instead a more general loss
function can be used. Thus we have:
C
(
x
,
t
)
2
C
−
D
(
x
,
t
)=
−
L
(
C
,
x
,
t
)
(4.3)
t
where the term on the right-hand side is the inactivation function. This function
will be composed of a global component for general, background NO reactions,
L
1
(
, representing structures which
act as local NO sinks such as blood vessels. The kinetics of these reactions are not
understood perfectly [47], but empirical data indicates either first or second order
decay [25, 27, 28, 43, 45], as represented by:
C
)
, and spatially localised components,
L
2
(
C
,
x
,
t
)
n
L
i
(
C
,
x
,
t
)=
k
i
(
x
)
×
C
(
x
,
t
)
(4.4)
where
n
is the order of the reaction and is equal to either 1 or 2, referred to as first or
second decay order respectively, and
k
i
is the reaction rate, commonly given in terms
of the half-life
t
1
/
2
=
k
i
. The reaction may also depend on the concentration
of the oxidative substance (usually oxygen), but, apart from very special cases (as in
[7], for example), this can be assumed to be constant and is left out of the equation
as it is subsumed by the reaction rate constant.
Values for the half-life of NO have been determined empirically and are depen-
dent on the chemical composition of the solvent within which NO is diffusing. For
instance, the half-life of NO in the presence of haemoglobin (Hb) is reported as be-
ing between about 1
ms
and 1
ln
(
2
)
/
s
, depending on the Hb concentration [23, 26, 28, 44].
In contrast, half-life values used for extravascular tissue, normally associated with
background NO consumption, are more than 1000 times longer, ranging from 1 to
>
µ
5
s
[29, 31, 43, 47].
The order of the reaction is also dependent on the nature of the diffusive environ-
ment. In environments where there is a high concentration of Hb, as in NO sinks,
recent work by Liu et al. [28] has shown that the reaction of NO with intact red
blood cells exhibits first order kinetics, which is in agreement with earlier measure-
ments [23, 26, 44]. Similarly, for modelling the global part of the loss function, as
would be seen in most extravascular regions of the brain, measurements of NO loss
are also consistent with first order decay [25, 28, 43]. Although second order de-
cay has been used for extravascular NO comsumption [27, 45], in these models NO
is diffusing
in vitro
in an air-saturated aqueous solution which is molecular-oxygen
rich (unlike intact extravascular brain tissue). Thus the dynamics of decay are taken
from empirical data in molecular-oxygen rich environs and are unlike those in the
intact brain. As we are concerned with modelling NO diffusion in the brain
in vivo
,
we have therefore used first order decay to model global NO loss in extravascular
tissue as well as in localised sinks. This gives us the following widely used equation
[24, 25, 26, 43, 44, 47] for diffusion of NO in the brain:
C
(
x
,
t
)
2
C
−
D
(
x
,
t
)=
−
k
(
x
)
C
(
x
,
t
)
(4.5)
t
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