Analysis of attenuation coefficient estimation in Fourier-domain OCT of semi-infinite media

Abstract

The attenuation coefficient (AC) is an optical property of tissue that can be estimated from optical coherence tomography (OCT) data. In this paper, we aim to estimate the AC accurately by compensating for the shape of the focused beam. For this, we propose a method to estimate the axial PSF model parameters and AC by fitting a model for an OCT signal in a homogenous sample to the recorded OCT signal. In addition, we employ numerical analysis to obtain the theoretical optimal precision of the estimated parameters for different experimental setups. Finally, the method is applied to OCT B-scans obtained from homogeneous samples. The numerical and experimental results show accurate estimations of the AC and the focus location when the focus is located inside the sample.

Fig. 1.

(a) The simulated single (blue) and averaged (red) OCT signals distorted by intensity-dependent Gaussian noise as a function of depth in physical distance. The averaged OCT signal was obtained by averaging over 500 single A-lines. (b) rCRLB values after averaging 1 to 1000 A-lines. The model parameter were set to $z_0$ = 160 µm, µ = 0.72$\text{m}{\text{m}}^{-1},C$ = 2.5×10⁴, and $z_R$ = 42 µm.

Fig. 2.

rCRLB matrices for models with intensity-dependent Gaussian noise. The model parameters were set to $z_0$ = 160 µm, µ = 0.72$\text{m}{\text{m}}^{-1},C$ = 2.5×10⁴, and $z_R$ = 42 µm.

Fig. 3.

The diagonal elements of the rCRLB matrices for the seven models (M1…M7) as shown in Table 1.

Fig. 4.

The error (%) of the estimated model parameters obtained from the diagonal elements of the rCRLB matrix for: a) $z_R$ = 42 µm, µ = 0.72 mm⁻¹, C = 2.5$\times {10}^4$ and $z_0$ = [−0.2,0.9] mm ; b) $z_0$ = 160 µm inside the sample, µ = 0.72 mm⁻¹, C = 2.5$\times {10}^4$ and $z_R$ = [0.01,0.5] mm; c) $z_R$ = 42 µm, $z_0$ = 160 µm inside the sample, C = 2.5$\times {10}^4$ and µ= [0.01,8] mm⁻¹ ; d) $z_R$ = 42 µm, $z_0$ = 160 µm, µ = 0.72 mm⁻¹ inside the sample and C = [${10}^2$,${10}^6$] (arb. units). The vertical dashed lines indicate the parameter values, which were set in the simulations and also were considered to be fixed for the other plots in this figure.

Fig. 5.

The coefficient of variation (CoV) (left column) and bias of the estimated parameters (right column) using the proposed method obtained from 100 simulated OCT signals, when initial parameter values are set to: a-b) 0 mm⁻¹≤ µ ≤ 6 mm⁻¹, $z_R=$50 µm, C = 2$\times$10⁴ (arb. unit), and $z_0=1\text{mm}$ ; c-d) $0\text{mm}\le z_0\le 2\text{mm}$, µ = 1 mm⁻¹, $z_R=$50 µm, and C = 2$\times$10⁴ (arb. unit); e-f) 0.01 mm ≤$z_R$ ≤ 0.2 mm, µ = 1 mm⁻¹, C = 2$\times$10⁴ (arb. unit), and $z_0=1$ mm, and g-h) 10³ ≤ C ≤ 7$\times$10⁴ (arb. unit), µ = 1 mm⁻¹, $z_R=$50 µm, and $z_0=1$ mm. The simulated model parameters were set to $z_R$ = 42 µm, C = 2.5$\times$10⁴ (arb. unit), µ = 0.72 mm⁻¹ and $z_0=0.16\text{mm}$.

Fig. 6.

Monte Carlo simulation results: a) the coefficient of variation (CoV) of the estimated parameters (solid lines) for 500 measurements and Cramer-Rao lower bounds (dashed lines); and (b) bias of the estimated parameters as a function of focus location inside the medium.

Fig. 7.

a) A B-scan of 0.05 wt. % TiO₂ in siliconewith adjusted focus location at 0.26 mm from the adjusted focus on the surface; b) The averaged A-lines (from the aquired B-scans per focus position) as a function of focus position; c) Averaged OCT signals (circles) along data-points located at 63 µm inside the sample (dashed lines in (a)) with the best fitted focus model.

Fig. 8.

The result of fitting the constrained model (blue) to the averaged A-lines per B-scan (red) obtained from the sample with 0.05 wt. % TiO₂ in silicone for eight different focus positions {−0.5, −0.3, −0.1, 0.15, 0.3, 0.45, 0.6, 0.75} mm from left to right. The location of the estimated focus (within the shown depth range) is indicated by the vertical dashed line.

Fig. 9.

The estimated model parameters as a function of focus position obtained averaged A-lines per B-scan acquired from the sample with 0.05 wt. % TiO₂ in silicone. The vertical red dashed lines indicate the B-scan in which the focus was on the sample’s surface.

Fig. 10.

Estimated model parameters: (a) z_R and (b) µ, obtained from the averaged A-lines of 2, 4, 8 and 16 B-Scans acquired at different focus positions inside the sample with 0.05 wt. % TiO₂ in silicone. The estimated and expected z₀ are shown for the combinations of (c) 2, (d) 4, (e) 8, and (f) 16 B-Scans, i.e. focus locations.