Biomedical Engineering Reference
In-Depth Information
measured value and the expected value can be evaluated with the standard deviation
s
or the
2 :
variance
s
N X
N
1
2
I i
I
s
¼
(8.30)
i
¼
1
where I i is the measured local value, I is the expected or average value, and N is the total number of
the samples taken. If a pixel is taken as a sample in the ROI, N is the number of pixels in the ROI. If the
sample is a subregion or an array of pixels, N is the number of subregions in the ROI. The size of the
subregion is called the scale of scrutiny and will be discussed later. For an unmixed binary system of
two separate components, the initial variance is
0 ¼ pð
s
1
(8.31)
where p is the proportion of one component. For instance, if the two initial components with initial
values I ¼
1 and I ¼
0 are mixed with a ratio of 1:1, the proportion of one component is
0 ¼
p
¼
1
1
þ
1
Þ¼
0
:
5 and the initial variance is
s
0
:
5
ð
1
0
:
5
Þ¼
0
:
25. If this system is fully
2
mixed, the fina l variance will be zero
0, because only one component with
expected value I exists at the end of an ideal mixing process.
In an actual mixing process, the uniformity of random distribution of the two components depends
on the size of the sample taken or the subregion. A variance close to zero can be accepted for full
mixedness. Mixing in microscale often involves gas and liquids only. Thus, different spatial scales
need to be considered in evaluating the mixedness: molecular size (nanometer scale), pixel size
(micrometer scale), and ROI scale (submillimeter scale). Considering the six orders of magnitude
difference between the molecular size and the pixel size, point uniformity is possible, e.g., each pixel
can achieve the final expected intensity value.
If the smallest size of mixed component is on the order of the pixel size, for instance, in the case of
solid particles, the pixel cannot be taken as the sample. In this case, an array of pixels or a subregion
should be taken for the analysis. The smallest size of the sample is called the scale of scrutiny. Since
a pixel is the smallest scale achievable with sampling by image microscopy, a pixel is a safe sample size
for liquids and gases in amicromixer. For gases and liquids, the pixel is six orders ofmagnitude larger than
the scale of scrutiny, which is the size of the molecules themselves. Mixing large particles would require
a subregion much larger than the particles, because the scale of scrutiny can be as large as many pixels.
In practical use, the degree of mixedness needs to be correlated with the mixing time, which is the
time needed to achieve full mixing. However, variance as a degree of mixedness does not correlate well
with time. If the mixing process is defined as the change of variance from
s
N ¼
0
ð
1
0
Þ¼
0 to
2
N
s
s
, the rate equation is
formulated for the variance as
d s
2
d t ¼
k s
2
2
N
s
(8.32)
where k is the rate of the mixing process. Integrating the above equation from the initial state ( t ¼
0,
2
s
0 ) to a given time t results in the solution of the dimensionless variance:
s
2
2
N
s
¼
exp
ð
kt
Þ
(8.33)
2
s
0 s
2
N
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