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10.3 Optical Microangiography (OMAG) 215
(a) (b)
4
32
1 mm
1 mm
Figure 10.9 The entire cerebrovascular flow
over the cortex of an adult mouse with the
skull intact was imaged with OMAG in vivo.
(a,b) The projection views of blood perfusion
before and after the right carotid artery was
blocked. (c) Photograph of the skull with the
skin folded aside, taken right after the exper-
iments. (d) Photograph showing blood ves-
sels over the cortex after the skull and the
meninges of the same mouse were carefully
removed. (Source: Images are reproduced
from Ref. [23].)
Figure 10.9 shows another OMAG imaging result obtained from an ischemic
thrombotic stroke experiment on a mouse model. Figure 10.9a shows the blood
flow in the cerebral cortex of a mouse with the skull intact. Occlusion of one carotid
artery does not cause cerebral infarction or neurological deficits in the mouse.
Figure 10.9b is taken from the same animal while the right carotid artery was
blocked for 5min. It is apparent that the blood flow in the cortex, rather than in the
right hemisphere only, is reduced as compared to Figure 10.9a. This is consistent
with the well-known collateral circulation between brain hemispheres, leading to
a total reduction in cortical blood volume due to the occlusion of one common
carotid artery. The capacity of OMAG to achieve such high-resolution imaging
of the cerebral cortex – within minutes and without the need for dye injections,
contrast agents, or surgical craniotomy, – makes it an extremely valuable tool for
studying the hemodynamics of the brain.
10.3.2.2 Directional Flow Imaging in OMAG
The base model of first-generation OMAG is capable of resolving 3D distribution of
dynamic blood perfusion at the capillary level within microcirculatory beds in vivo.
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216 10 Optical Microangiography: Theory and Application
However, this base method does not give the capability to image the directional
flow, which seriously limits the applications for OMAG, because the blood flow
information regarding its direction is often important in a number of investigations,
for example, in the study of complex flow dynamics in the microfluidic mixers and
in the investigation of blood flow involvement in cerebrovascular diseases such
as ischemia, hemorrhage, vascular dementia, traumatic brain injury (TBI), and
seizure disorders. To overcome this drawback, Wang [65] proposed a method that
forces the RM to move back and forth. However, this would significantly increase
imaging complexity. For example, the OMAG imaging speed is reduced by half
and the computational load on OMAG is doubled to obtain meaningful blood
flow images, because OMAG needs to acquire two three-dimensional volumetric
spectrogram data sets. It was then quickly realized that a digital modulation
frequency (DMF) method can be used to manipulate the modulated interferograms
so that the directional flow information can be obtained from one 3D volumetric
spectrogram data set [66], without resorting to the hardware approach in [65].
Following the OMAG method described in the earlier section, assume the
frequency modulation in the interferograms is f
M
, which is a constant frequency.
Here f
M
can be provided by a number of approaches, for example, moving the RM
at a constant velocity in one direction [23, 60, 65] or offsetting the sample beam at
the scanner that gives the B-scan image [67]. For simplicity, the real function of a
spectral interferogram can be expressed by Eq. (10.12). If we construct the analytic
function of Eq. (10.12) by performing the Hilbert transform in terms of t
1
(note that
t
2
is constant), then this analytic function is always Eq. (10.13), because f
0
is always
>0, which is guaranteed by placing the sample surface below the zero-delay line. In
this case, if we digitally multiply Eq. (10.13) with a complex function exp(−j4π f
M
t
2
)
and then take the real part of the results, we can simply transfer Eq. (10.12) into
B
(
t
1
, t
2
)
= cos
2πf
0
t
1
− 2π
f
M
+ f
D
t
2
+ ϕ
(10.18)
It is feasible because f
M
is known a priori. Now, we can construct the analytic
function of Eq. (10.18) by considering t
2
as a variable, while t
1
to be constant. Then,
if f
M
+ f
D
< 0, the analytic function is
˜
H
(
t
1
, t
2
)
= cos
2πf
0
t
1
− 2π(f
M
+ f
D
)t
2
+ ϕ
+ j sin
2πf
0
t
1
−2π (f
M
+ f
D
)t
2
+ ϕ
(10.19)
but if f
M
+ f
D
>
0, the analytic function becomes
˜
H
(
t
1
, t
2
)
= cos
2πf
0
t
1
− 2π(f
M
+ f
D
)t
2
+ ϕ
− j sin
2πf
0
t
1
−2π (f
M
+ f
D
)t
2
+ ϕ
(10.20)
Consequently, Eqs. (10.18–10.20) simulate the exact situation of the mirror moving
in a direction opposite to that as described in Section 10.3.1. This derivation of
the analytic function provides a solid mathematical basis for obtaining information
on the directions of blood flow from only one 3D spectrogram data set captured
by OMAG. As a result, the directional blood flow imaging can be obtained by the
following approach: the analysis detailed in the previous section is used to give one
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10.3 Optical Microangiography (OMAG) 217
No flow signal
Flow signal
Flow signal
No flow signal
(a)
(
b
)
(c)
(
d
)
Figure 10.10 (a,b) The OMAG structural and flow
images calculated from the spectrograms captured when
f
M
= 400 Hz. (c,d) Corresponding results computed from
the digital frequency modulation (DFM) approach applied to
the same data set. Scale bar = 500 µm. (Source: Images are
reproduced from Ref. [66].)
image of blood flow direction when the RM moves toward the incident beam, and
the analysis described here is used to give an image of blood flow in the opposite
direction, as if the mirror moved away from the incident beam.
Figure 10.10 shows the imaging results obtained through the OMAG method
and DMF approach. Two microtubes, within which scattering particles are flowing
with flow directions against each other, were buried under a scattering phantom.
Figure 10.10a,b are the microstructure and flow images respectively, which were
directly obtained from the OMAGalgorithm. Applying the DMF approach, different
direction flow and microstructure images are present in Figure 10.10c,d. It is shown
that the directional flow images can be successfully achieved by OMAG and DMF
algorithms.
10.3.3
Second-Generation OMAG
In the first-generation OMAG, as explained in the earlier section, the moving and
static scatters were separated by introducing a carrier frequency, and this carrier
frequency is generated by adding a phase delay to each A-scan using a moving
reference arm or an off-pivot scanning beam. However, these techniques rely
on precise synchronization of reference arm modulation and B-scan acquisition,
which require expensive or cumbersome modulators, and are unable to detect
bidirectional flow in a single B-scan. In order to overcome this drawback, a
modified Hilbert transform without the use of spatial frequency modulation is
introduced in the OMAG. No frequency modulations are required, and a new
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218 10 Optical Microangiography: Theory and Application
OMAG signal-processing algorithm based on a modified Hilbert transform is
applied to the spatial frequency content across a single B-scan to obtain volumetric
perfusion map with directional information. The new second-generation OMAG is
applicable for both spectrometer-based OCT and SS-OCT systems, provided they
have comparable B-scan acquisition rates. Unlike previously described techniques
[23, 66], which require two separate B-scans to detect positive and negative flow,
each with modulations tuned to the desired flow direction, the second-generation
OMAG is able to resolve bidirectional flow in a single B-scan pass across the
sample.
10.3.3.1 Theoretical Overview
If we consider the OMAG imaging of blood flow within the microcirculation tissue
beds, under a B-scan condition, the static scatterers and the moving scatterers are
illustrated in Figure 10.11a. The depth-encoded spectral interferogram captured by
the linear detector is a real function, B(z, t) and can be written as
B(z, t) = A(z)cos(k
0
z) (10.21)
where A(z) is the reflective coefficient of the scatterer, k
0
is the wave vector of the
detected light, and z relates to the detecting depth. In order to simplify the problem,
we do not consider the random phase arising from the refractive index float and
assume that the reflective coefficient A(z) of the scatter is a constant along the
B-scan direction; t is the B-scan time corresponding to the different lateral position.
According to Eq. (10.21), if we consider the nonmoving scatterers at a depth
z, as shown in Figure 10.11a,b, the signal B(z, t) does not vary with the B-scan
time, t. So the corresponding signal can be viewed as a rectangular function,
that is, rect
(
t/τ
0
)
= F
(
t −τ
0
/2
)
− F
(
t +τ
0
/2
)
, as shown in Figure 10.12a. On the
other, for the moving scatterer, the signal is a rectangular enveloped cosine signal,
that is, rect
(
t/τ
0
)
cos
= cos
(
2πu
0
t
)
F
(
t −τ
0
/2
)
− F
(
t +τ
0
/2
)
, which is illustrated
Depth z
B-scan direction
(a)
(b)
(c)
Static scatterer
Moving scatterer
t
0
2
A(z)cos(k
0
z)
t
0
T
0
t
0
N
0
n
0
A(z)cos(k
0
z + 2pn
0
t)
t
0
+
t
0
2
t
0
−
Figure 10.11 (a– c) Schematic of moving and nonmoving
scattering structure under B-scan condition in SD-OCT.
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10.3 Optical Microangiography (OMAG) 219
B-scan line
600550500450400
0
0.5
1
600550500450400
0
0.5
1
−0.5
−1
Frequency number
−500 −250
250 500
0
0
0.01
0.02
−500 −250
250 500
0
0
0.04
0.02
BW
(a)
(b)
(
c
)
(d) Frequency number
B-scan line
Figure 10.12 Comparison of structural signal and flow
signal with N
0
and N
0
= 1000. (a) The structural signal,
(b) the absolute FFT of the structural signal, (c) the flow
signal, and (d) the absolute FFT of the flow signal.
in Figure 10.12c, where u
0
is the modulation frequency generated by the flow
speed.
10.3.3.1.1 Frequency Domain Representation of Static Scatterer
If we consider the above assumption, according to Figure 10.11b,c, the total time
required for a single B-scan is T
0
, which corresponds to a total of N
0
A-lines in a
B-scan frame. The scanning time for the scatterer (moving or static), τ
0
, depends on
the number of A-lines in the B-frame, n
0
, to image the structure, which corresponds
to the physical dimension of the imaged structure within a B-scan frame, and t
0
is the B-scan time required at the center of the scattering structure as illustrated
in Figure 10.11c. According to the above parameters, the FT R(z, u) of the lateral
dimension (i.e., B-scan at time t) can be shown as
R(z, u) =
A(z)cos
k
0
z
T
0
t
0
−
τ
0
2
t
0
−
τ
0
2
e
−i2πut
dt (10.22)
where u is the FT frequency of B-scan at time t. This corresponds to the spatial
frequency along the B-scan direction. After simplification, the result of Eq. (10.22)
can be presented as
R(z, u) = A(z)cos
k
0
z
sin
(
πuτ
0
)
πuT
0
e
−i2πut
0
(10.23)
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220 10 Optical Microangiography: Theory and Application
If we consider the OMAG spectral detection section, the acquisition time of the
linear detector is 1/f
0
(sampling rate), that is, the time required to acquire each
A-scan in a B-scan frame. Then the total acquisition time T
0
required for one B-scan
can be expressed as: T
0
= N
0
/f
0
. The B-scan time for the scattering structure is
τ
0
= n
0
/f
0
. In frequency domain, the total frequency is f
0
and the frequency step
is f
0
/N
0
. The variable frequency u can be expressed as u = nf
0
/N
0
, where n is the
frequency number. Since the spectral signal detected by the camera is a real signal
and therefore its FT is Hermitian, it has both positive and negative frequency
components. Then Eq. (10.22) can be modified and represented as
R
±
(z, n) = A(z)cos
k
0
z
n
0
N
0
sin c
n
0
n
N
0
e
−i2πnt
0
f
0
N
0
(10.24)
where ± denotes the positive and negative frequencies. In Eq. (10.24), the range
of the nonmoving scatterer, that is, n
0
/N
0
determines the oscillation frequency
and the amplitude of sin c function. The depth-encoded information is retained
in cos
k
0
z
, which is independent of the FT along the B-scan direction. Owing
to the Hermitian symmetry of the FT, the frequency ranges from −f
0
/2tof
0
/2;
accordingly, the frequency number n ranges from −N
0
/2toN
0
/2, with the max
value of the sin c function sin c(x) centered at n = 0. This means that the main
signal energy concentrates at the zero frequency. Figure 10.12a shows the absolute
frequency signal of the nonmoving structural (static) signal. For example, we
set the total B-scan lines N
0
to 1000, and the total B-scan lines of nonmoving
scatterer n
0
to 50 (between 475 and 525). The reflectivity of the signal is assumed
to unity, that is, B(z, t) = 1. The absolute frequency signal
R
±
(z, n)
is shown
in Figure 10.12b; the amplitude of
R
±
(z, n)
has a maximum value of 0.05
(n
0
/N
0
= 0.05), which is located at the zeroth frequency position with a frequency
span of n = 2N
0
/n
0
, and the other consequent peaks on the either sides of the
zeroth peak have a frequency span of n = N
0
/n
0
. This is because the biological
tissue is often optically heterogeneous in nature, which means the magnitude of
the backscattered light and the refractive index are functions of time variable. So the
intensity captured by the linear detector will be modulated by the heterogeneous
properties of the sample along each B-scan. The spatial frequency components of
a static tissue sample, which we call the heterogeneous frequencies, will exhibit as
a randomly distributed function around zero frequency with a bandwidth of BW,
which is indicated in Figure 10.12b. This spatial frequency bandwidth depends on
the heterogeneity of the static scatterers and is called heterogeneous frequency.
10.3.3.1.2 Frequency Domain Representation of Moving Scatterers
For the moving scatterers under the B-scan condition, the corresponding schematic
is shown in Figure 10.11c. The depth-encoded real spectral interference signal
recorded by the SD-OCT system can be written as
B(z, t) = A(z)cos
k
0
z +2π ut
(10.25)
where t is the B-scan time corresponding to the lateral position across B-scan
direction, u is the modulation frequency generated by the flow speed, where
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10.3 Optical Microangiography (OMAG) 221
u = 2u
0
cos θ
d
/λ and u
0
is the absolute velocity of moving scatterers, and λ is
the wavelength of detected light. Here, the flow speed u can be viewed as a
carrier frequency, and the flow associated with the moving scatterers include the
negative flow (scatterers moving away from the sample beam) and the positive flow
(scatterers moving forward to the sample beam). Then the FT of the spectral signal
B(z, t) can be expressed as
H(z, u) =
A(z)
T
0
t
0
+
τ
0
2
t
0
−
τ
0
2
cos
k
0
z +2π ut
e
−i2πu
0
t
dt
=
A(z)e
ik
0
z
2T
0
t
0
+
τ
0
2
t
0
−
τ
0
2
e
−i2π
(
u−u
0
)
t
dt +
A(z)e
−ik
0
z
2T
0
t
0
+
τ
0
2
t
0
−
τ
0
2
e
−i2π
(
u+u
0
)
t
dt
(10.26)
This equation includes positive and negative frequency components
H
(
z, u
)
= H
+
(
z, u
)
+ H
−
(
z, u
)
(10.27)
where H
+
(
z, u
)
is the positive frequency and H
−
(
z, u
)
is the negative frequency.
Then Eq. (10.26) can be split up as
H
+
(
z, u
)
=
A(z)e
ik
0
z
2T
0
sin
[
πτ
0
(
u −u
0
)
]
π
(
u −u
0
)
e
−i2π
(
u−u
0
)
t
0
(10.28)
H
−
(
z, u
)
=
A(z)e
−ik
0
z
2T
0
sin
[
πτ
0
(
u +u
0
)
]
π
(
u +u
0
)
e
−i2π
(
u+u
0
)
t
0
(10.29)
According to the assumption described in Section 10.2.1, Eq. (10.8) can also be
expressed as
H
+
(
z, n
)
= A(z)e
ik
0
z
n
0
2N
0
sin c
n
0
N
0
n −
u
0
N
0
f
0
e
−i2π
nf
0
N
0
−u
0
t
0
(10.30)
H
−
(
z, n
)
= A(z)e
−ik
0
z
n
0
2N
0
sin c
n
0
N
0
n +
u
0
N
0
f
0
e
−i2π
nf
0
N
0
+u
0
t
0
(10.31)
If we consider Eqs. ((10.28) –(10.31)) with Eqs. (10.23) and (10.24), the nonmov-
ing scatterers are centered at the zeroth frequency (DC) region (heterogeneous
frequency) and the moving scatterers are shifted away from the zeroth frequency
n = 0ton =±u
0
N
0
/f
0
. This shifting distance in the frequency domain can be
directly related to the velocity u
0
of the moving scatterers, total B-scan lines, N
0
, and
the acquisition rate of CCD, f
0
. Since the max frequency number is n
max
=±N
0
/2,
the maximum detectable velocity with OMAG is v
max
=±λf
0
/4, which is the
same as that with DOCT. Figure 10.12c,d illustrates a simulated signal for the
moving scatterer and the corresponding frequency shift for the signal in the fre-
quency domain, respectively. The two shifted sin c function peaks in Figure 10.12d
correspond to Eqs. (10.28) and (10.29). The simulated illustration in Figure 10.12c,d
clearly shows how the OMAG separates the moving scatterers from nonmoving
scatterers by flow speed modulation.
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222 10 Optical Microangiography: Theory and Application
100 200 300 400 500
0
0.01
0.02
Frequency number
100 200 300 400 500
0
0.01
0.02
Frequency number(b)
(a)
Figure 10.13 (a) The flow frequency signal with v
0
= 2
mm s
−1
and shift distance n = 101. (b) The flow frequency
signal with v
0
= 6mms
−1
and the shift distance n = 302.
10.3.3.1.3 Parameters That Influence the Flow Signal Detection
Flow Velocity
The frequency shift inducedby the flow modulationcan directly relate
to the flow velocity of the moving scatterers, which is illustrated in Figure 10.13a,b.
For example, we consider a B-scan condition at λ = 840 nm, with a sampling
frequency f
0
= 47kHz and sampling density of, N
0
= 1000 (A-lines in a B-scan) for
a moving scatterer that covers A-lines of n
0
= 50. Figure 10.13a shows the signal
in positive-frequency space, where the flow velocity v
0
= 2mms
−1
by assuming
that the Doppler angle, θ
d
= 0. Based on Eqs. (10.29) and (10.30), the shifting
frequency number can be calculated by n =±2ν
0
N
0
/λf
0
= 101, which is shown in
Figure 10.13a. In case of velocity v
0
= 6mms
−1
, the shifting frequency number
is n = 302, which is shown in Figure 10.13b. As the flow speed increases, the
frequency shift also increases, and at low flow speed, it is hard to separate moving
scatterers from the static scatterer.
Size of the Moving Scatterer For a specific scan density, for example, N
0
= 1000, the
size of the moving scatterer also influences the detected flow signal. For example,
100 200 300 400 500
0
0.005
0.01
Frequenc
y
number
100 200 300 400 500
0
0.01
0.02
Frequenc
y
number(a) (b)
Figure 10.14 (a) The flow frequency signal with the flow
size n
0
= 15 and (b) the flow frequency signal with the flow
size n
0
= 50.
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10.3 Optical Microangiography (OMAG) 223
50 100 150 200 250
0
0.01
0.02
Frequency number
200 400 600 800 1000
0
0.01
0.02
Frequency number(a) (b)
Figure 10.15 (a) The flow frequency signal with the total
B-scan line N
0
= 500 and (b) the flow frequency signal with
the total B-scan line N
0
= 2000.
we consider a moving scatterer with a flow size (oscillation cycle) of T = N
0
/n
0
with a span of 2N
0
/n
0
. Figure 10.14a,b shows the flow signals with a size of n
0
= 15
and n
0
= 50, respectively, for a flow velocity of v
0
= 2.5mms
−1
, the imaging
parameters are the same as mentioned above. It is obvious that the span of the first
peak in Figure 10.14a is broader than that of the first peak in Figure 10.14b.
Sampling Line Density The influence of scan density N
0
along the lateral scan
direction (B-scan density) on the flow signal is shown in Figure 10.15. If we
consider a B-scan with a sampling rate of f
0
= 47 kHz and a moving scatterer with
velocity v
0
= 2.5mms
−1
and size n
0
/N
0
= 0.025, the flow signal detected with
scan densities N
0
= 500 and N
0
= 2000 are shown in Figure 10.15a,b, respectively.
For a moving scatterer, high B-scan density shifts the flow signal away from the
zero frequency regions more than low B-scan density does. Thus, by increasing
the B-scan density, it is possible to effectively separate flow signal from structural
signal, that is, the heterogeneous frequency noise should be reduced. On the other
hand, the imaging speed would be sacrificed.
10.3.3.1.4 Flow Extraction in OMAG
A schematic of the OMAG signal processing flow chart is illustrated in Figure 10.16.
Figure 10.16a shows a B-scan frame, which includes the static and moving scatter-
ers. According to Eqs. (10.21) and (10.25), if we consider the nonmoving scatterers
at a depth z, the signal B(z,t)does not vary with the B-scan time t. Therefore, the
corresponding signal can be viewed as a rectangular function.On the contrary, for
the moving scatterers, the signal under B-scan condition is a rectangular envelope
cosine signal, which is shown in Figure 10.16a. By applying a Fourier transforma-
tion along the B-scan direction, the spatial frequency components of a static tissue
sample, which we call the heterogeneous frequencies, will exhibit as a randomly
distributed function around zero frequency with a bandwidth of BW, which is
indicated in Figure 10.16b. On the other hand, the moving scatterers produce a
frequency shift, which is caused by the Doppler effect of the moving particles,
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224 10 Optical Microangiography: Theory and Application
400 600 800 1000 1200
−1
−0.5
0
0.5
1
B-scan line
−3000 −1500 0 1500 3000
0
0.01
0.02
0.03
0.04
0.05
0.06
Frequenc
y
number
400 600 800 1000 1200
0
0.2
0.4
0.6
0.8
1
1.2
B-scan line
−3000 −1500 0 1500 3000
0
0.02
0.04
0.06
Frequency number
FFT
IFFT
(a) (b)
(c)(d)
Figure 10.16 The flow chart of signal processing steps in
OMAG. (a) The flow signal and structural signal, (b) the
flow signal and structural signal in the frequency domain,
(c) filtering of the structural signal and negative frequency
of the flow signal, and (d) the flow signal inverting from the
frequency domain.
and shifts it away from the heterogeneous frequencies due to static scatterers.
This is illustrated in Figure 10.13b. Next, for extracting the flow information, the
structural signal is filtered out in the frequency domain. This was accomplished by
introducing the modified Hilbert transform algorithm. To achieve this, one of the
conjugate terms was windowed out and the heterogeneous frequency was centered
at the zeroth frequency region (DC) using a frequency-shifted Heaviside function.
The cutoff frequency shift of the Heaviside function depends on the heterogeneity
of the static scatterers. Then, an inverse FT is applied; the inverse FT has positive
and negative frequency. After inverse transforming from frequency domain, and
then applying the conventional spectral FT along the depth direction, the strength
of the flow signal will be retrieved. This is illustrated in Figure 10.16c,d. For the
flow signal, the inverse FT has positive and negative frequency, which retrieves the
directional flow information and can be expressed as
iFFT
(
H
+
(
z, n
))
=
A(z)e
i
(
k
0
z+2πu
0
t
)
2
(10.32)
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