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Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT)

Nate J. Kemp, Haitham N. Zaatari, Jesung Park, H. Grady Rylander III, and Thomas E. Milner

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Abstract

Form-biattenuance (Δχ) in biological tissue arises from anisotropic light scattering by regularly oriented cylindrical fibers and results in a differential attenuation (diattenuation) of light amplitudes polarized parallel and perpendicular to the fiber axis (eigenpolarizations). Form-biattenuance is complimentary to form-birefringence (Δn) which results in a differential delay (phase retardation) between eigenpolarizations. We justify the terminology and motivate the theoretical basis for form-biattenuance in depth-resolved polarimetry. A technique to noninvasively and accurately quantify form-biattenuance which employs a polarization-sensitive optical coherence tomography (PS-OCT) instrument in combination with an enhanced sensitivity algorithm is demonstrated on ex vivo rat tail tendon (mean Δχ=5.3·10-4, N=111), rat Achilles tendon (Δχ=1.3·10-4, N=45), chicken drumstick tendon (Δχ=2.1·10-4, N=57), and in vivo primate retinal nerve fiber layer (Δχ=0.18·10-4, N=6). A physical model is formulated to calculate the contributions of Δχ and Δn to polarimetric transformations in anisotropic media.

©2005 Optical Society of America

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Supplementary Material (4)

Media 1: MOV (1188 KB)     
Media 2: MOV (1144 KB)     
Media 3: MOV (1184 KB)     
Media 4: MOV (1272 KB)     

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Figures (5)
Fig. 1.
Fig. 1. Noise-free model polarization arc [P(z), black] and eigen-axis ( β ^ , green) on the Poincaré sphere (left) and corresponding normalized Stokes parameters [Q(z), U(z), V(z)] vs. depth (right). Polarizations at the front [P(0)] and rear [P(Δz)] specimen surfaces are represented by red and blue dots respectively. (a) Pure form-birefringence causes rotation of P(z) around β ^ in plane Π1 which is normal to β ^ . (b) Pure form-biattenuance causes translation of P(z) toward β ^ in plane Π2. (c) Combined birefringence and biattenuance cause P(z) to spiral toward β ^ and orthogonal planes Π1 and Π2 are therefore functions of depth [Π1(z) and Π2(z)]. Movies showing 3D nature of Poincaré sphere: a (1.21 MB); b (1.17 MB); c (1.21 MB).

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Fig. 2.
Fig. 2. Depth-resolved polarization data [S 1(z), orange] and associated noise-free model polarization arc [P 1(z), black] and eigen-axis ( β ^ , green) determined by the multistate nonlinear algorithm in rat tail tendon are shown on the Poincaré sphere (left). Corresponding normalized Stokes parameters [Q(z), U(z), V(z)] and associated nonlinear fits (black) are shown on the right. A single incident polarization state (m=1) is shown for simplicity. (a) S m (z) for tendon with relatively high form-biattenuance (Δχ=8.0·10-4) collapses toward β ^ faster than that for (b) tendon with relatively low form-biattenuance (Δχ=3.0·10-4).

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Fig. 3.
Fig. 3. S1(z) (orange) and associated P 1(z) (black) and β ^ (green) determined by the multistate nonlinear algorithm in rat Achilles tendon are shown on the Poincaré sphere (left). A single incident polarization state (m=1) is shown for simplicity. Form-biattenuance in this specimen (Δχ=3.2 °/100µm) is lower than for specimens shown in Figures 2(a) and 2(b) and spiral collapse toward β ^ is correspondingly slower.

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Fig. 4.
Fig. 4. S m (z) (colored) and associated P m (z) (black) and β ^ (green) for in vivo primate RNFL shown on the Poincaré sphere for M=6 (left). Corresponding normalized Stokes parameters [Q(z), U(z), V(z)] and associated nonlinear fits (black) are shown for a single incident polarization state (m=1, right). Notice the RNFL exhibits only a fraction of a wave of phase retardation compared to multiple waves exhibited by tendon specimens in Figures 2(a), 2(b), and 6. Movie showing 3D nature of Poincaré sphere (1.30 MB).

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Fig. 5.
Fig. 5. A model for form-biattenuance consisting of alternating anisotropic and isotropic layers.

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Equations (28)

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J = [ exp ( ( Δ χ + i Δ n ) π Δ z ⁄ λ 0 ) 0 0 exp ( ( − Δ χ − i Δ n ) π Δ z ⁄ λ 0 ) ]
= [ ∣ ξ 1 ∣ exp ( i arg ( ξ 1 ) ) 0 0 ∣ ξ 2 ∣ exp ( i arg ( ξ 2 ) ) ] ,
J = [ ∣ ξ 1 ∣ exp ( i δ ⁄ 2 ) 0 0 ∣ ξ 2 ∣ exp ( − i δ ⁄ 2 ) ] .
D = ∣ T 1 − T 2 ∣ T 1 + T 2 = ∣ ∣ ξ 1 ∣ 2 − ∣ ξ 2 ∣ 2 ∣ ∣ ξ 1 ∣ 2 + ∣ ξ 2 ∣ 2 0 ≤ D ≤ 1 ,
Δ n = λ 0 2 π δ Δ z = n s − n f ,
Δ χ = χ s − χ f ,
ε = 2 π λ 0 Δ z Δ χ ,
J = [ exp ( ε + i δ 2 ) 0 0 exp ( − ε − i δ 2 ) ] ,
D = ∣ e ε − e − ε ∣ e ε + e − ε = tanh ( ε ) .
d P ( z ) dz + ( P ( z ) × β ⇀ re ) + P ( z ) × ( P ( z ) × β ⇀ im ) = 0 ,
β ⇀ = β ⇀ re + i β ⇀ im = ( β re + i β im ) β ̂ .
β = β re + i β im = 2 π λ 0 ( Δ n + i Δ χ ) ,
2 δ = 2 β re Δ z ,
γ ( z ) = 2 tan − 1 [ tan ( γ ( 0 ) 2 ) exp ( − 2 β im z ) ] 0 ≤ γ < π ,
2 ε = 2 β im Δ z .
PSNR = l arc σ speckle ,
d l arc = 2 ( β re 2 + β im 2 ) 1 2 sin [ γ ( z ) ] dz ,
l arc = [ 1 + ( δ ε ) 2 ] 1 2 [ γ ( 0 ) − γ ( Δ z ) ] .
l arc ≈ 2 ( δ 2 + ε 2 ) 1 2 sin [ γ ( 0 ) ] .
S m ( z ) = ( Q ( z ) U ( z ) V ( z ) ) = ( 〈 E h , m ( z ) 2 − E v , m ( z ) 2 〉 N A 〈 2 E h , m ( z ) E v , m ( z ) cos [ Δ ϕ c , m ( z ) ] 〉 N A 〈 2 E h , m ( z ) E v , m ( z ) sin [ Δ ϕ c , m ( z ) ] 〉 N A ) ⁄ 〈 E h , m ( z ) 2 + E v , m ( z ) 2 〉 N A .
W m ( z ) = 〈 ( E h , m ( z ) 2 − E v , m ( z ) 2 2 E h , m ( z ) E v , m ( z ) cos [ Δ ϕ c , m ( z ) ] 2 E h , m ( z ) E v , m ( z ) sin [ Δ ϕ c , m ( z ) ] ) ⁄ E h , m ( z ) 2 + E v , m ( z ) 2 〉 N A .
R M = ∑ m = 1 M R o [ S m ( z j ) , W m ( z j ) ; 2 ε , 2 δ , β ̂ , P m ( 0 ) ] ,
R o = ∑ j = 1 J { W ( z j ) [ S ( z j ) − P [ z j ; 2 ε , 2 δ , β ̂ , P ( 0 ) ] } 2 ,
n p 2 = ( h 1 a ) n f 2 + ( 1 − h 1 a ) n w 2 and
n s 2 = n w 2 + ( h 1 ⁄ a ) ( n f 2 − n w 2 ) 1 + 1 2 ( 1 − h 1 a ) ( n f 2 − n w 2 n w 2 ) .
Δ n = h 1 h ( n p − n s ) .
t p t s ( z ) = [ n p ( n s + n w ) 2 n s ( n p + n w ) 2 ] z h ,
Δ χ = − λ 0 2 π h ln ( n p ( n s + n w ) 2 n s ( n p + n w ) 2 ) .
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