Achieving High-Precision Attenuation Coefficient Measurement in Optical Coherence Tomography

Abstract

In this study, we aim to validate the analytical Cramer-Rao lower bound (CRLB) equation for determining attenuation coefficients using a 1310 nm Optical Coherence Tomography (OCT) system. Our experimental results successfully confirm the validity of the equation, achieving unprecedented precision with a standard deviation below 0.01 mm^-1 for intralipid samples. Furthermore, we introduce a systematic framework for attaining high precision in OCT attenuation measurements.

Keywords: Cramér–Rao lower bound; Fisher matrix; attenuation coefficient; curve‐fitting; depth‐resolved estimation; optical coherence tomography; precision.

FIGURE 1

Method 1—Sequential outline of data processing and analysis.

FIGURE 2

Method 2—Sequential outline of data processing and analysis.

FIGURE 3

Roll‐off data (blue dots) and its 10th‐order polynomial fit (orange line), normalized to 1. The plot demonstrates a gradual decrease in intensity up to 1.9 mm, followed by a sharper decline thereafter.

FIGURE 4

Analysis for deriving confocal parameters. (a, b) display CRLB and standard deviation from the four‐parameter fit (Equation 1) to average A‐scans (N = 12 500). A 0.2% concentration volume was chosen for its precision and proximity to theoretical bounds. (c) shows normalized residuals, confirming a stable fit ($R^2$ = 0.997). $z_f$ is at 0.9250 ± 0.0005 mm with a $z_R$ of 0.3516 ± 0.0013 mm. (d) illustrates the resulting normalized point spread function, with peak shift to 1.6 mm due to refractive index correction n = 1.34.

FIGURE 5

Average A‐scans (N = 12 500) after correction for roll‐off, point spread function, and noise offset, shown for a selected range (0.2%, 1%, 5.7%, and 22.7%) of intralipid dilutions to maintain clear visibility. Blue curves represent two‐parameter fits, Equation (3), in dB scale. The corrected A‐scans show expected linear profile post‐correction, transitioning to exponential decay in linear scale. Please note that the width of the fitting range $\left|\mathit{AFR}\right|$ decreases with concentration to assure a SNR above 20 dB. Residual offset persists post‐noise correction, and at lower concentrations, exponential decay does not fully reach noise floor observed in higher concentrations. System artifacts are visible at depth of 2.4 mm for the highest concentrations.

FIGURE 6

Method 1: Comparison of ${\mu }_{\mathit{OCT}}$ (blue dots) and ${\sigma }_{\mathit{OCT}}$ (red dots) obtained from the two‐parameter fit, Equation (3) to the N = 12 500 average A‐scans. The orange dots represent the analytical CRLB, Equation (6), indicating the theoretical best precision achievable ${\sigma }_{{\mu }_{\mathit{OCT}}}$. The dashed line represents the linear regression of the first five concentration volumes, resulting in an intercept of ${\mu }_a=0.08m{m}^{-1}$.

FIGURE 7

Plot of the measured ${\sigma }_{{\mu }_{\mathit{OCT}}}$ as a function of the expected ${\sigma }_{{\mu }_{\mathit{OCT},\mathit{\text{an}}}}$ based on the CRLB, Equation (6), for all IL concentrations and a range of averages using Method 1. The linear regression, depicted by the dashed black line, shows a slope of 0.9795 ± 0.0008 and an intercept of 0.0002 ± 0.0002 $\mathrm{m}{\mathrm{m}}^{-1}$. The goodness of fit, $R^2$ is 0.979.

FIGURE 8

Comparison of experimental values and predictions. (a) Comparison between experimental ${\mu }_s$ (depicted as blue dots), and predicted ${\mu }_s$ (Equation 7) using parameters a = 1.868 $\mathrm{n}{\mathrm{m}}^{-1}$ and b = 0.0005481 $\mathrm{m}{\mathrm{m}}^{-1}$ (represented by the red curve), and with adjusted parameters a = 1.6 and b = 0.00063 (orange line). (b) Direct comparison illustrating the parameter shift, where an overestimation of the theoretical predicted ${\mu }_s$ from Aernouts et al. ([21], red dot), compared to a shift of the parameter values a and b (orange dots), becomes evident.

FIGURE A1

Averaged A‐scans (N = 12 500) following corrections for roll‐off, point spread function, and noise offset. The blue curves correspond to two‐parameter fits based on Equation (3), presented on a dB scale. The corrected A‐scans exhibit the anticipated linear decay profile after correction, transitioning to an exponential decay when plotted on a linear scale. It should be noted that the width of the fitting range (∣AFR∣) decreases with increasing concentration to maintain an SNR above 20 dB. Despite noise correction, a residual offset remains. At lower concentrations, the exponential decay does not fully reach the noise floor observed in higher concentrations. System artifacts are apparent at a depth of 2.4 mm for the highest intralipid concentrations.

FIGURE B1

Method 2: Comparison of ${\mu }_{\mathit{OCT}}$ (blue dots) and ${\sigma }_{\mathit{OCT}}$ (red dots) obtained from the two‐parameter fit, Equation (3). The orange dots represent the theoretical best precision (CRLB). An attenuation value of 0.19 $m{m}^{-1}$, extracted for the 0.2% intralipid concentration (${\mu }_{\mathit{OCT},0.2}$) from Method 1, has been added as an offset to the results obtained from Method 2.

FIGURE B2

Plot of the measured ${\sigma }_{{\mu }_{\mathit{OCT}}}$ as a function of the expected ${\sigma }_{{\mu }_{\mathit{OCT},\mathit{\text{an}}}}$ based on the CRLB, Equation (6), for all IL concentrations and a range of averages, using Method 2. The linear regression, depicted by the dashed black line, shows a slope of 0.9802 ± 0.0070 and an intercept of 0.0005 ± 0.0001 mm. The goodness of fit, $R^2$ is 0.98.

FIGURE C1

The illustration of residuals for the 22.7% intralipid dilution demonstrates a stable fit (R ² = 1.00) using model Equation (3). This result confirms that our assumption of single scattering is adequate.

FIGURE D1

Illustration of the experimentally determined standard deviation (red dots) extracted from the two‐parameter fit, Equation (3), and the analytical lower bound (orange dots), Equation (6), as a function of the number of A‐scans taken for averaging, using Method 1. The error bars depict the range of variations in both analytical and experimental precision across different concentrations.

FIGURE D2

Comparison of the experimentally determined standard deviation (red dots), obtained from the two‐parameter fit (Equation 3), and the analytical CRLB (orange dots), calculated using Equation (6). Both are plotted as a function of the number of A‐scans used for averaging, based on Method 2. The error bars depict the range of variations in both analytical and experimental precision across different concentrations.