Biomedical Engineering Reference
In-Depth Information
can be used, it is inappropriate for our needs due to the asymptotically slow conver-
gence. Due to the accuracy requirements and relative smoothness of the function,
the approach we have taken is to use successive applications of one-dimensional in-
tegration. In this method, to evaluate
y
(
,
t
i
)
x
for each abscissa,
t
i
, of an iteration of
the outer integration of
y
(
,
)
dxdt
, we must perform one whole numerical in-
tegration over
x
using that value of
t
i
to evaluate
y
x
t
at each
x
j
. This means
that if it takes
N
function evaluations to get a sufficiently accurate estimate for the
one-dimensional integral, we will need around
N
2
evaluations to achieve the same
accuracy for the two-dimensional integral. Moreover, since it is good practice for ac-
curacy reasons to use simple numerical integration routines, the exponential growth
in the number of operations needed is exacerbated by the slow convergence of these
methods. Given these considerations, solutions for reasonably complicated func-
tions requiring greater than double integration is probably best handled with one of
the numerical methods discussed in the next section.
For the results detailed here, the extended trapezoidal rule given in Equation 4.23
[37] was used for the outer integration (over time) with the inner integration (over ra-
dial distance) performed by the 'quad8' function to speed up convergence. However,
to ensure accuracy, we checked the solutions by performing both inner and outer in-
tegrations using the extended trapezoidal rule. In evaluating the continuous solution
for the tubular source, it should be noted that at
t
(
x
j
,
t
i
)
=
0 the instantaneous solution is:
Q
for
R
1
<
r
<
R
2
ρ
/
=
=
C
A
(
R
1
,
R
2
,
r
,
t
)=
Q
2 for
r
R
1
,
r
R
2
(4.31)
0
else
Solutions were accurate to a relative accuracy of 0
5%. Accuracy of solutions for the
tubular and spherical sources were further checked using the numerical integration
package in the programming language Maple which is very accurate and accounts
for improper integrals correctly, but is too slow for general use.
.
4.2.3
Modelling diffusion of NO from an irregular 3D structure
4.2.3.1
Finite difference methods for diffusive problems
From the previous section it is clear that the analytical method is not tractable for
many situations which we might want to investigate. In particular, modelling irreg-
ularly shaped sources and sinks is inappropriate and other numerical techniques for
solving the partial differential equations (PDEs) governing the spread of NO must be
used. For diffusive problems evolving over the short time-scales associated with NO
diffusion in the brain, one recommended approach is to use finite differences [1, 37].
These methods proceed by approximating the continuous derivatives at a point by
difference quotients over a small interval, for instance replacing
∂
x
t
by:
x
x
(
t
+
∆
t
)
−
x
(
t
)
t
=
(4.32)
t
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