Biomedical Engineering Reference
In-Depth Information
the less smooth areas of the integrand near the temporal origin. This means that the
integration procedure must be modified for cases where the lower limit of integration
in Equation 4.19 is less than 1
ms
to include the following analytical approximation
for the second part of the integral:
Dt
π
exp
(
exp
(
dt
ε
2
2
t
1
r
a
−
r
)
a
+
r
)
e
−
λ
Q
−
(4.24)
4
Dt
4
Dt
0
h
h
1
e
−
λ ε
2
2
2
+
(
r
−
a
)
(
r
+
a
)
Q
,
r
,
ε
−
,
r
,
ε
(4.25)
4
D
4
D
where:
r
Dt
,
ε
)=
ε
0
k
t
dt
1
e
−
h
(
k
,
r
e
−
3
r
D
k
3
erfc
k
ε
2
√
(4.26)
ε
√
ε
(
ε
−
k
2
=
2
k
)+
erfc
(
x
)=
1
−
erf
(
x
)
(4.27)
and noting that at
t
=
0 the instantaneous solution is:
Q
for
r
<
a
C
S
(
a
,
r
,
t
)=
Q
ρ
/
2 for
r
=
a
(4.28)
0
else
In the above
a
is the radius of the sphere,
r
is the distance from its centre and
1
ms
.
Solutions for the hollow sphere and when the lower limit of integration is greater than
zero can be derived from the above equations.
The approximation in Equation 4.24 is based on the principle that if
f min
[
0
,
ε
]
ε
≤
is
the minimum value attained by a function
f
(
t
)
over the range
[
0
,
ε
]
and
f max
[
0
,
ε
]
is
the maximum of
f
(
t
)
over the same range then:
ε
ε
f min
[
0
,
ε
]
+
f max
[
0
,
ε
]
f
(
t
)
g
(
t
)
dt
g
(
t
)
dt
(4.29)
2
0
0
which has a maximum error of:
ε
f max
[
0
,
ε
]
−
f min
[
0
,
ε
]
g
(
t
)
dt
(4.30)
2
0
Thus, the actual value of the error is dependent on the parameter values used but, for
the parameters used here, the errors are small enough to keep the solutions to within
a relative accuracy of 0
1%.
The continuous solution for the tubular source (Equation 4.22) has to be inte-
grated over both space and time, necessitating a slightly different approach since
multi-dimensional integration is significantly more time-consuming and can mag-
nify errors and instabilities in the methods used [37]. While Monte-Carlo integration
.
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