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Separation of absorption and scattering profiles in spectroscopic optical coherence tomography using a least-squares algorithm

Chenyang Xu, Daniel L. Marks, Minh N. Do, and Stephen A. Boppart

Open Access Open Access

Abstract

In spectroscopic optical coherence tomography, it is important and useful to separately estimate the absorption and the scattering properties of tissue. In this paper, we propose a least-squares fitting algorithm to separate absorption and scattering profiles when near-infrared absorbing dyes are used. The algorithm utilizes the broadband Ti:sapphire laser spectrum together with joint time-frequency analysis. Noise contribution to the final estimation was analyzed using simulation. The validity of our algorithm was demonstrated using both single-layer and multi-layer tissue phantoms.

©2004 Optical Society of America

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Figures (6)
Fig. 1.
Fig. 1. Errors in retrieved absorption fa as a function of SNR in SOCT I(λ, z).

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Fig. 2.
Fig. 2. Emission spectrum of the Ti:Al2O3 laser (black curve) and absorption spectrum of dye ADS830WS (red curve). Also shown is the FWHM region used for determining W(λ).

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Fig. 3.
Fig. 3. Absorption/scattering attenuation loss due to various absorbers/scatterers as measured by a spectrometer: 40 μM ADS830WS NIR dye solution, 1% 160 nm silica microbead solution, 0.5% 330 nm silica microbead solution, and 0.5 % 800 nm silica microbead solution, potato slice, and murine skin.

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Fig. 4.
Fig. 4. Separation of absorption and scattering losses from distinctive interfaces: total attenuation profile measured by SOCT (black curve), resolved absorption profile by the separation algorithm (green curve), resolved scattering profile by the separation algorithm (red curve), the sum of the resolved absorption profile and the resolved scattering profile (blue curve).

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Fig. 5.
Fig. 5. Resolved absorber and scatterer concentrations in turbid media. (a) Resolved cumulative dye concentrations from solutions with different dye concentrations but the same microbead concentration. (b) Resolved cumulative microbead concentrations from solutions with different microbead concentrations but the same dye concentration. The smooth lines represent the least-squares-fitted model of Eq. (10). The resolved concentrations retrieved from the slopes of the fitted lines are shown as well. The insets show the diagrams of the samples.

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Fig. 6.
Fig. 6. Resolved absorber and scatter concentrations in a turbid multi-layer phantom. (a) Resolved cumulative dye concentration. (b) Resolved cumulative microbead concentration. The lines represent the least-squares-fitted model of Eq. (10). The resolved concentrations retrieved from the slopes of the fitted lines are also shown. The inset in (a) shows the diagram of the samples.

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Equations (24)

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I ( λ , z ) = S ( λ ) H r ( λ , z ) H m ( λ , z ) H s ( λ , z ) .
H m ( λ , z ) = exp ⁡ { − 2 ∫ 0 z [ μ a ( λ , z ′ ) + μ s ( λ , z ′ ) ] dz ′ } .
H r ( λ , z ) = H r ( λ ) H r ( z ) ,
H s ( λ , z ) = H s ( λ ) H s ( z ) .
I ( λ , z ) = S ( λ ) H s ( λ ) H r ( λ ) R ( z ) exp ⁡ [ − 2 ∫ 0 z [ μ a ( λ , z ′ ) + μ s ( λ , z ′ ) ] dz ′ ] ,
I ( λ , z = 0 ) = S ( λ ) H r ( λ ) H s ( λ ) R ( z = 0 ) .
I ′ ( λ , z ) ≜ I ( λ , z ) I ( λ , z = 0 ) = R ′ ( z ) exp ⁡ { − 2 ∫ 0 z [ μ a ( λ , z ′ ) + μ s ( λ , z ′ ) ] dz ′ } .
μ a ( λ , z ) = ε a ( λ ) f a ( z ) ,
μ s ( λ , z ) = ε s ( λ ) f s ( z ) ,
− 2 ∫ 0 z [ μ a ( λ , z ′ ) + μ s ( λ , z ′ ) ] dz ′ = − 2 [ ε a ( λ ) ∫ 0 z f a ( z ′ ) dz ′ + ε s ( λ ) ∫ 0 z f s ( z ′ ) dz ′ ] .
Y ( λ , z ) ≜ log ⁡ [ I ′ ( λ , z ) ]
= log ⁡ R ′ ( z ) − 2 [ ε a ( λ ) ∫ 0 z f a ( z ′ ) dz ′ + ε s ( λ ) ∫ 0 z f s ( z ′ ) dz ′ ]
= − ε a ( λ ) F a ( z ) − ε s ( λ ) F s ( z ) + C ( z ) .
F a ( z ) ≜ 2 ∫ 0 z f a ( z ′ ) dz ′ , F s ( z ) ≜ 2 ∫ 0 z f s ( z ′ ) dz ′ , C ( z ) ≜ log ⁡ R ′ ( z ) .
E ( z ) = ∫ λ 1 λ 2 [ − F a ( z ) ε a ( λ ) − F s ( z ) ε s ( λ ) + C ( z ) − Y ( λ , z ) ] 2 W ( λ ) d λ ,
[ ∫ λ 1 λ 2 ε a 2 ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 ε s ( λ ) ε a ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 ε a ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 ε a ( λ ) ε s W ( λ ) d λ ∫ λ 1 λ 2 ε s 2 ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 ε s ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 ε a ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 ε s ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 W ( λ ) d λ ] ︸ A [ − F a ( z ) − F s ( z ) C ( z ) ] ︸ X = [ ∫ λ 1 λ 2 Y ( λ , z ) ε a ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 Y ( λ , z ) ε s ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 Y ( λ , z ) W ( λ ) d λ ] ︸ Y .
[ ∫ λ 1 λ 2 ε a 2 W ( λ ) d λ ∫ λ 1 λ 2 λ ε s ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 ε a ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 λ ε a ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 λ 2 W ( λ ) d λ ∫ λ 1 λ 2 λ W ( λ ) d λ ∫ λ 1 λ 2 ε a ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 λ W ( λ ) d λ ∫ λ 1 λ 2 W ( λ ) d λ ] [ − F a ( z ) − a F s ( z ) D ( z ) ] = [ ∫ λ 1 λ 2 Y ( λ , z ) ε a ( λ ) W ( λ ) d λ ∫ λ 1 λ 2 λ Y ( λ , z ) W ( λ ) d λ ∫ λ 1 λ 2 Y ( λ , z ) W ( λ ) d λ ] ,
AX = Y ,
X α = arg min { ∥ Y − AX ∥ 2 + α 2 ∥ LX ∥ 2 } .
X = ( A T A + α 2 L T L ) − 1 A T Y .
X = A − 1 Y .
σ x i 2 = ∑ j = 1 3 b ij 2 σ y j 2 ,
F = λ LD [ F LD ( F high − F low ) + F low ] λ ( F high − F low ) + F low
δ λ = λ c 2 2 l STFT ,
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