3.1.

Depth-Resolved Estimation

Vermeer et al.⁸ introduce the model as shown in Eq. (4), where $I[z]$ is the OCT signal of a pixel and $\mu [z]$ is the OAC, both at depth $z$, and $\mathrm{\Delta }$ is the pixel size (usually related to the axial resolution of the OCT system):

Another problem is that the DRE method has not taken the OCT noise floor into account. The noise floor includes electrical noise of the photodetector, shot noise, and relative intensity noise from the reference arm light.¹⁸ If the OCT signal is fully attenuated when it reaches the depth $z_S$, then all of the depths from $z_S+1$ to $N$ may be noise. The DRE method adds a large amount of noise to the denominator of Eq. (5), and the resulting OAC may be smaller than it should be. In this paper, we propose the ODRE method to solve these problems.

3.2.

Optimized Depth-Resolved Estimation

The DRE method is suitable for the boundary condition $I(\infty )=0$, so Eq. (4) can be used for subsequent calculations and the OAC for the last point $N$ should be expressed as

With this equation, we can supplement the data after the last point $N$, because $\sum _{i=z+1}^{\infty }I[i]$ can be split into $\sum _{i=z+1}^{N}I[i]$ and $\sum _{i=N+1}^{\infty }I[i]$. Equation (4) can be rewritten as

Fig. 1

OCT signal cropping. The black curve is a typical average OCT signal. The blue curve indicates noise floor measured with the reference light only. The red curve represents the position that is 5 dB above the noise floor. The small black square is where the signal is cropped. All the curves were obtained by averaging 100 consecutive measurements to facilitate comparison.

Fig. 2

The relationship between the pixel number and the OAC. All values reported as the mean ± S. E. M; (*) represents $p<0.05$ and (**) represents $p<0.01$.

Fig. 3

Exponential fitting is performed for the end of the imaging depth (120 pixels). The solid blue line is a typical averaged A-scan OCT signal. The solid red line represents the result of the fitting.

4.1.

Numerical Simulations

Reliable testing and simulation play an important role in the innovation of optical imaging, measurement techniques, and data analysis algorithms. To evaluate the effectiveness of the ODRE method experimentally, a series of numerical simulations were performed with and without noise. We first simulated a beam of light propagating through a homogeneous medium without noise and using three sets of OACs (0.5, 0.7, 1). The observed depth was 3 mm, and the pixel size Δ was 0.01 mm. As shown in Fig. 4(a), for ease of calculation, we assumed that the initial light intensity is 1. The OACs of the simulated data were estimated by the method of DRE and ODRE, respectively. For ODRE method, the number of pixels used for exponential fitting is 100 (1 mm). The results are shown in Figs. 4(b) and 4(c).

Fig. 4

Numerical simulation without noise. (a) Simulated OCT signals. (b) DRE results; the small graph to the right of (b) shows the depth from 2.5 to 3 mm, indicating a huge error. (c) ODRE results.

Generally, the OACs at different depths in a homogeneous medium should be the same, but there is increasing error in the results of DRE and its maximum is $\mu [N-1]=I[N-1]/2\mathrm{\Delta }I[N]\approx \frac{1}{2\mathrm{\Delta }}=50{\mathrm{mm}}^{-1}$. It should be noted that the estimation error of the DRE method exists at any depth and not just at the end of the data. On the other hand, OACs obtained by the ODRE method are satisfactory and consistent with the preset OACs (0.5, 0.7, and 1).

As mentioned above, DRE method can only be used if almost all light is attenuated within the recorded imaging depth range. In practice, the OCT data does not necessarily satisfy such a depth range. As a result, an unacceptable error is generated, when the DRE method is used. Here, we selected three depth ranges of 0.5, 1, and 1.5 mm, respectively. The shallower the depth, the more light is left [as shown in Fig. 5(a)]. The OACs were estimated within these limited depth ranges by DRE and ODRE. The results are shown in Figs. 5(b) and 5(c), respectively.

Fig. 5

Numerical simulation for homogeneous media with finite depth (a) simulated OCT signals, the preset OAC is $0.5{\mathrm{mm}}^{-1}$, (b) DRE results, and (c) ODRE results (the error is only 0.5%).

As shown in Fig. 5, a major error can occur if the optical signal attenuation is not completed within the limited depth range. The smaller the depth range, the sooner the error occurs. Instead, the ODRE method can be used to estimate the OACs in any depth range with negligible error.

OCT images are often affected by speckle noise, which is a multiplicative noise of the coherent imaging system.^20,21 To accurately reflect the performance of the proposed method in OAC estimation in a noisy environment, three sets of numerical signals were set up, and each set was actually a combination of multiplicative noises and the corresponding analog datasets used in Fig. 4. The increasing process of the noise follows the development:

Fig. 6

Numerical simulation with random noise. (a) Simulated OCT signals. (b) DRE results; the small graph to the right of (b) shows the depth from 2.5 to 3 mm. (c) ODRE results.

Figure 6(a) shows the original simulated data, and Figs. 6(b) and 6(c) show the estimated value of the OACs obtained by the DRE and the ODRE, respectively.

According to Fig. 6, this series of results are similar to that of Fig. 4. Again, the results of ODRE are still consistent with the preset OACs, but the difference is that the former fluctuates by noise. These convincing simulation results show that the proposed method can accurately estimate the OACs in a random depth regardless of the noise.

4.2.

Phantom Experimental Results

Four liquid optical phantoms comprising different Intralipid concentrations were prepared. The volume-to-volume (v/v) ratios used were $\sim 5\%$, 10%, 15%, and 20% (diluted in distilled water). Three methods (exponential fitting, DRE, and ODRE) were used to estimate the OAC. For the DRE method, the entire imaging depth was used for OAC estimation. For the ODRE method, only 0.5-mm depth range was used. The number of pixels used for exponential fitting is 60 (0.15 mm). Figure 7 shows the experimental results of 20% phantom. Figure 7(a) shows the OCT intensity image, Fig. 7(b) shows the corresponding OAC image obtained by DRE using Eq. (4), and Fig. 7(c) shows the corresponding OAC image obtained by ODRE using Eq. (8).

Fig. 7

(a) OCT intensity image from uniform phantoms; (b) and (c) corresponding OAC image obtained by DRE and ODRE, $\mathrm{\Delta }\approx 0.0025\mathrm{mm}$. (d) A typical OCT A-scan within (a) and the result of exponential curve fitting. The result ($\mu$) is $\sim 3.0{\mathrm{mm}}^{-1}$. (e) The blue curve is the estimated result obtained by the ODRE method. The red dotted line is the corresponding average OAC at the depth range of 0.5 mm, which is consistent with the exponential fitting result. The cyan curve is the estimated result obtained by the DRE method; the OAC value is underestimated due to the accumulation of excessive noise during the estimation process. The orange dotted line is the corresponding average OAC, which is significantly smaller than the result of exponential fitting.

Since the phantoms are homogeneous, the exponential fitting method can be applied to the entire depth range, and the results are credible and can be used as a standard for evaluating the other methods.⁸ The blue solid line in Fig. 7(d) is shown as the optical signal data for the leftmost column (A-scan) of Fig. 7(a). The exponential fitting method is used to process the data, and the result is shown in the red solid line in Fig. 7(d). The result ($\mu$) is $\sim 3.0{\mathrm{mm}}^{-1}$. Figure 7(e) is the results of ODRE and DRE. It can be seen that the OAC curve within the depth range obtained by the ODRE method is even, and the average OAC value (red dotted line) is consistent with the exponential fitting result. The OAC value obtained by the DRE method is underestimated due to the accumulation of excessive noise during the estimation process. The OAC of each phantom obtained from the exponential fitting, DRE, and ODRE are listed in Table 1.

Table 1

OAC estimated by the exponential fitting, DRE, and ODRE methods from the uniform phantoms.

It can be seen from Table 1 that the OAC obtained by the DRE method is generally small, and the error is larger with the increase in the concentration. This is because the faster the signal attenuate, the greater is the noise accumulated by the DRE method on the denominator of Eq. (5) and the greater is the error.

To better validate the capabilities of ODRE in practical applications, a multilayer phantom was created. The phantom consisted of three thin layers (15%, 5%, and 10%), 0.25 mm per layer. Each layer is separated by a 0.12-mm coverslip. The multilayer phantom has a total depth of 1.2 mm. The number of pixels used for exponential fitting is 60 (0.15 mm).

The experimental results are shown in Fig. 7.

Figure 8(a) shows the OCT intensity image of the multilayer phantom, A∼E represent different layers. Among them, A, C, and E are 15%, 5%, and 10% of phantom, respectively. Here, B and D are the coverslips. Figure 8(b) shows the corresponding OAC image obtained by ODRE. Figure 8(c) is a typical average A-scan within Fig 8(a), Fig. 8(d) is the OAC corresponding to Fig. 8(c). The OAC results for the three-layer phantom are $\mu (A)=2.04\pm 0.20$, $\mu (C)=0.83\pm 0.26$, and $\mu (E)=1.20\pm 0.25$, respectively. A $T$-test is performed between the OACs obtained by the ODRE from multilayered samples and the OAC obtained by the exponential curve-fitting from homogenous sample. There is no significant difference for layer A (15%), and there is a significant difference for layer C (5%) and E (10%), $p<0.05$. This may be due to a lower SNR at a deeper depth.

Fig. 8

Estimated OCA for the multilayer phantom by ODRE. (a) OCT intensity image; A, C, and E are 15%, 5%, and 10% of phantom, respectively. B and D are the coverslips. (b) Corresponding OAC image obtained by ODRE. (c) A typical A-scan within (a); (d) corresponding OAC of (c).

4.3.

In Vivo Mouse Brain Imaging

About 3-month-old C57/616 mice (20 to 30 g) were used in the study. All procedures were performed in accordance with the Animal Ethics and Administrative Council of Northeastern University. All efforts were made to minimize animal suffering and to reduce the number of animals used. Surgical anesthesia was induced with sodium pentobarbital (3%, 5 mg/100 g, IP). The scalp was cut along the midline of the skull, and the interparietal bone was exposed by pulling the skin to the side of the head. A standard $3\times 3{\mathrm{mm}}^2$ cranial window, covering the distal branches of MCA and the anterior cerebral artery (ACA), was created in the left parietal cortex, 0.5-mm lateral from the sagittal suture and 0.5-mm posterior from the bregma. A circular coverslip was placed on the exposed brain surface and sealed on bone with a dental cement.²⁴ After surgery, the mice were placed into the sample arm of the OCT system and prepared for baseline data acquisition.

The model of the focal cerebral ischemia used in our experiments was based on the previously described endothelin-1-induced middle cerebral artery occlusion (MCAO) model.¹² The stereotaxic coordinates of the ET-1 injection were $-0.83\text{-}\mathrm{mm}$ anterior, $-3.36\text{-}\mathrm{mm}$ lateral, and 1.0-mm ventral relative to bregma. A dose of $3\text{-}\mu \mathrm{L}$ ET-1 (dilute to 100 pmol/L in 0.9% normal saline) was injected at $1\mu \mathrm{L}/\min$. After injection, we waited for 3 min to ensure that the drug was completely absorbed. The needle was slowly removed, and the mouse was immediately placed under the sample arm to begin the temporal scans. During date acquisition, the mouse was under gas anesthesia (isoflurane) instead of IP anesthesia.

In mouse brain imaging, all the datasets were taken at the depth of 1 mm to avoid the noise floor. The number of pixels used for exponential fitting is 120 (0.3 mm). In Fig. 9(a), the orange box represents the ROI in our experiment, highlighted with a black arrow. As shown in Fig. 9(b), the baseline en face OCT angiogram, up to the depth of $150\mu \mathrm{m}$, was obtained from the ROI, and yellow arrows indicate arteries whereas blue indicate veins. Figure 9(c) shows the en face OAC image obtained from the same ROI by the ODRE method. The attenuation coefficient of the ischemic region, but not the healthy region, increases with the development of ischemia so that they can be distinguished during the process. The en face OAC images were reconstructed by averaging OACs at a specific depth range (151 to $300\mu \mathrm{m}$) because there is a better contrast between the damaged and the healthy tissues at this range.

Fig. 9

(a) Scan of a subject’s skull; the ROI for data acquisition is indicated by a black arrow; the red dot indicated by the white arrow is the position of the ET-1 injection. (b) The baseline en face OCT angiogram, up to $150\text{-}\mu \mathrm{m}$ depth. (c) Average en face OAC plots with a depth range of 151 to $300\mu \mathrm{m}$, with the OACs in the range of 0.3 to 3 (${\mathrm{mm}}^{-1}$). (d) The results of OCT angiograms and OAC images of the mouse cortex during ET-1-induced MCAO, respectively; the white bar represents $200\mu \mathrm{m}$.

The OCT angiography provides the detailed information on cerebral blood flow, whereas the OAC mapping provides information on the optical properties of the tissue. These two techniques can be combined into an indicator to show the status of brain tissue. When the brain tissue is damaged by ischemia, blood perfusion changed first, and OAC changed after a period of time (20 to 40 min). Figure 9(d) shows the changes in cerebral perfusion and OAC of the cerebral cortex in mice with focal cerebral ischemia over time.

As shown in Fig. 9(d), when ET-1 was injected, due to a high concentration of ET-1, the arteries and most capillaries near the injection point (a red dot) were blocked rapidly and the blood supply reached a minimum level within 10 min. However, the arteries far from the injection site were not obstructed, which may be due to some compensatory mechanisms, such as anastomosis of MCA and ACA.²⁴ The veins also remained unchanged. Part of the reason might be that their vascular walls are relatively thin and the walls are almost unaffected by ET-1. During the period from 20 to 120 min, arteries recovered gradually, as indicated by the yellow arrows in Fig. 9(d), with part of the capillaries also returning to normal.

As shown in Fig. 9(d), the white area gradually appeared in the OAC images over time, indicating that the backscattering of light increased and reflected that the optical properties of the tissue have changed. The white region was located near the blocked artery, suggesting that tissue in this area has been infarcted due to insufficient supply of blood and oxygen.

In the ischemic region, lesions did not occur immediately at the onset of occlusion, and the lesion appeared and the size of the lesion area increased over time. In the later stage, after the effect induced by ET-1 disappeared, blood vessels began to recover gradually. However, we can see that the injured brain cells have not recovered, which indicates that the damage to the brain cells is irreversible. This result is consistent with the conclusion of the reference.⁹

To investigate the effects of cerebral ischemia on OAC in more detail, we studied the OAC of brain tissue in damaged and undamaged areas. The results are shown in Fig. 10.

Fig. 10

(a) OCT angiogram at 40 min with the MCA end on the left and the ACA end on the right; the white lines represent the cross sections of the tissue for the OCT structure images (c)–(e) and the OCT angiograms (f)–(h). (b) OAC image at 40 min; the yellow line indicates the cross sections of the tissue for the OAC images (i)–(k) and average OAC curve (l)–(n); the red box 1 and the blue box 2 are the sampling locations chosen representatively as the injured area and the noninjured area, respectively; both region sizes are $10\times 10\text{pixels}$. (o) The time-varying curve of the OAC in regions 1 and 2, with all values reported as the mean ± S. E. M. (p) and (q) OAC curves varying with depth in regions 1 and 2, respectively.

Figure 10(a) shows a 40-min OCT angiogram, and Fig. 10(b) shows a corresponding time OAC image. The white line in Fig. 10(a) and the yellow line in Fig. 10(b) represent the same cross-sectional location that passes through the damaged cerebral region. The OCT structural images [Figs. 10(c)–10(e)], OCT angiograms [Figs. 10(f)–10(h)], and OAC images [Figs. 10(i)–10(k)] all are at this cross-sectional location.

From the OCT structural images, we can see that the penetration depth of the OCT light (green box) at baseline is significantly greater than that at 40 and 120 min. Penetration depth refers to the depth at which the signal intensity is reduced to 5% of its initial value. The penetration depth is $>1\mathrm{mm}$ at baseline and $<0.5\mathrm{mm}$ at 40 and 120 min. This indicates that the attenuation of light in the damaged cerebral tissue is enhanced. In the OCT angiograms, the blood vessels are abundant at baseline, and they are minimal at 40 min and recover at 120 min. It is not easy to identify changes in OAC in the damaged region from the cross-sectional images, so we chose a specific depth range of 151 to $300\mu \mathrm{m}$ (yellow box) and plotted the average OAC curve in that range. Figures 10(l)–10(n) are the average OAC curves for the baseline, 40 min, and 120 min, respectively. From these curves, it is noticeable that the average OACs across the 151- to $300\text{-}\mu \mathrm{m}$ depth of the cross-section are approximately the same at the baseline. In contrast, at 40 and 120 min after ischemia, the OACs in the middle part (injured region) are significantly higher than those on both sides of the image.

The red and blue curves in Fig. 10(o), respectively, correspond to the damaged region (red region 1) and the nondamaged region (blue region 2) shown in Fig. 10(b). The OAC curve for region 1 significantly increases after ischemia, peaks at 40 min, and then fluctuates slightly. The curve for region 2 does not change noticeably before 50 min, after which the OAC gradually increases and the gap between it and the red curve narrows. This indicates that the brain tissue in region 2 was also affected by ischemia, although it was located relatively far from the ET-1 injection site. Figures 10(p) and 10(q) are a series of OAC curves for a typical A-scan of region 1 and region 2, respectively, all of which vary with depth. Different colors and shapes are intended to represent different times. The above results show that OAC can be used as an indicator for brain injury.