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Time-domain Raman analytical forward solvers

Fabrizio Martelli, Tiziano Binzoni, Sanathana Konugolu Venkata Sekar, Andrea Farina, Stefano Cavalieri, and Antonio Pifferi

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Abstract

A set of time-domain analytical forward solvers for Raman signals detected from homogeneous diffusive media is presented. The time-domain solvers have been developed for two geometries: the parallelepiped and the finite cylinder. The potential presence of a background fluorescence emission, contaminating the Raman signal, has also been taken into account. All the solvers have been obtained as solutions of the time dependent diffusion equation. The validation of the solvers has been performed by means of comparisons with the results of “gold standard” Monte Carlo simulations. These forward solvers provide an accurate tool to explore the information content encoded in the time-resolved Raman measurements.

© 2016 Optical Society of America

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Figures (7)
Fig. 1
Fig. 1 Diagram of a photon path trajectory in a x, z plane projection from the source (red circle) to the Raman scatter (dashed line) and from the Raman scatter to the detector (continuous line). Green circles are Tyndall interactions. The yellow circle is a Raman interaction.

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Fig. 2
Fig. 2 Schematic of the parallelepiped.

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Fig. 3
Fig. 3 Comparison for the TR Raman reflectance between analytical formulae and MC in absence (a, e, g) and in presence (c) of background fluorescence. For all the figure panels μ′sb = μ′sbe = 1, μabe = 7 · 10−3 mm−1, μsR = 10−3 mm−1, ni = nie = 1.4, no = noe = 1, g = 0. Specifically: (a) μab = 5 · 10−3 mm−1, μaf = 0, ρ = 20 mm; (c) μab = 5 · 10−3 mm−1, μaf = 7 · 10−3 mm−1, ηe=1 and τ = 2 ns, ρ = 20 mm; (e) μab = 5 · 10−3 mm−1, μaf = 0, ρ = 10 mm; (g) μab = 10−2 mm−1, μaf = 0, ρ = 20 mm.

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Fig. 4
Fig. 4 Dependence of the temporal profile of Raman photons for different values of μabe and μ′sb. The common parameters were ρ = 10 mm, ni = nie = 1.4, no = noe = 1 and μsR = 10−6 mm−1, μab = 0.01 mm−1. Specifically: (a) μabe ∈ [0, 0.035] mm−1 and μ′sb = μ′sbe = 1mm−1 and; (b) μ′sb = μ′sbe ∈ [0.5, 2] mm−1 and μabe = 0.01 mm−1.

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Fig. 5
Fig. 5 Comparison of the temporal profile of the DE Raman signal (Eq. (20)), and of the heuristic model (Eq. (12)). The common parameters were: ρ = 10 mm, ni = nie = 1.4, no = noe = 1, μsR = 10−6 mm−1, μ′sb = μ′sbe = 1mm−1, and μab = 0.01 mm−1.

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Fig. 6
Fig. 6 Relative per cent error of the heuristic model (Eq. (12)) as compared to the DE solver (Eq. (20)). The common parameters were ρ = 10 mm, ni = nie = 1.4, no = noe = 1, μsR = 10−6 mm−1, μab = 0.01 mm−1 and μ′sb = 1mm−1. Specifically: (a) μabe ∈ [0, 0.02] mm−1 and μ′sbe = 1mm−1 and; (b) μ′sbe ∈ [0.5, 1] mm−1 and μabe = 0.01 mm−1.

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Fig. 7
Fig. 7 Schematic of a diffusive cylinder.

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Tables (1)

Tables Icon

Table 1 Summary of the notations used for the optical properties of the medium at λ and λe in the case of a pure Raman model (column Raman) or a hybrid Raman and fluorescence model (column Raman+Fluorescence). The subscript b denotes the background, the subscript R denotes the Raman scattering and the subscript f denotes the fluorescence. In this work, we have assumed μsRe = 0 and μafe = 0. The optical parameters appearing in column Raman have to be used with Eqs. (18), (20) and (39), while optical parameters appearing in column Raman+Fluorescence have to be used with Eqs. (24), (28) and (32).

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Equations (39)

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[ 1 v ∂ ∂ t + μ a − D ∇ 2 ] Φ ( r , t ) = q ( r , t ) ,
[ 1 v e ∂ ∂ t + μ a e − D e ∇ 2 ] Φ e ( r , t ) = q e ( r , t ) ,
q e ( r ′ , t ′ , λ e ) = μ s R Φ ( r s , r ′ , μ a , μ s ′ , n i , t ′ , λ ) .
q e ( r ′ , t ′ , λ e ) = μ s R G ( r s , r ′ , μ a , μ s ′ , n i , t ′ , λ ) .
[ 1 v e ∂ ∂ t + μ a e − D e ∇ 2 ] Φ e ( r , t ) − μ s R G ( r s , r , t ) = 0 .
Φ e ( r , t ) = ∫ 0 t ∫ V ′ q e ( r ′ , t ′ ) G e ( r ′ , r , μ a e , μ s e ′ , n i e , t − t ′ ) d V ′ d t ′ ,
Φ e ( r , t ) = μ s R ∫ 0 t ∫ V ′ G ( r s , r ′ , μ a , μ s ′ , n i , t ′ ) G e ( r ′ , r , μ a e , μ s e ′ , n i e , t − t ′ ) d V ′ d t ′ .
Φ ( r , μ a = μ a b + μ s R , μ s ′ , t , λ ) = Φ ( r , μ a = μ a b , μ s ′ , t , λ ) exp ( − μ s R v t ) .
Φ e ( r , μ a e = μ a b e , μ s e ′ = μ s b e ′ , t , λ e ) = Φ e ( r , μ a b , μ s b ′ , t , λ e ) .
Φ e ( r , μ a e , μ s e ′ , t , λ e ) = Φ ( r , μ a b , μ s b ′ , t , λ ) − Φ ( r , μ a b , μ s b ′ , t , λ ) exp ( − μ s R v t ) ,
Φ e ( r , μ a e , μ s e ′ , t , λ e ) ≈ Φ ( r , μ a b , μ s b ′ , t , λ ) μ s R v t .
R eHeur ( ρ , μ a e , μ s e ′ , t , λ e ) = R ( ρ , μ a b , μ s b ′ , t , λ ) μ s R v t ,
G ( r , r ′ , t ) = 2 3 v L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K l x ′ ) cos ( K m y ) cos ( K m y ′ ) sin [ K n ( z + 2 A D ) ] sin [ K n ( z ′ + 2 A D ) ] exp [ − ( K l 2 + K m 2 + K n 2 ) D v ( t − t ′ ) − μ a v ( t − t ′ ) ] ,
L x ′ = L x + 4 A D ,     L y ′ = L y + 4 A D ,     L z ′ = L z + 4 A D ,
K l = ( 2 l − 1 ) π L x ′ ,         K m = ( 2 m − 1 ) π L y ′ ,     K n = n π L z ′ ;     l ,   m ,     n = 1 , 2 , 3 , 4 , 5 , …
Φ eRaman ( r , t ) = 2 3 μ s R v v e L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K m y ) sin [ K n ( z + 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e − D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e − μ a v ) ] − 1 × { exp [ − ( K l 2 + K m 2 + K n 2 ) D v t − μ a v t ] − exp [ − ( K l 2 + K m 2 + K n 2 ) D e v e t − μ a e v e t ] } .
R eRamanEPBC ( x , y , t ) = Φ eRaman ( x , y , z = 0 , t ) 2 A e .
R eRamanEBPC ( x , y , t ) = 2 2 μ s R v v e A e L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K m y ) sin [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e − D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e − μ a V ) ] − 1 × { exp [ − ( K l 2 + K m 2 + K n 2 ) D v t − μ a v t ] − exp [ − ( K l 2 + K m 2 + K n 2 ) D e v e t − μ a e v e t ] } .
R eRamanFick ( x , y , t ) = D e ∂ Φ eRaman ( x , y , z = 0 , t ) ∂ z .
R eRamanFick ( x , y , t ) = 2 3 D e μ s R v v e L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K m y ) K n cos [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e − D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e − μ a v ) ] − 1 × { exp [ − ( K l 2 + K m 2 + K n 2 ) D v t − μ a v t ] − exp [ − ( K l 2 + K m 2 + K n 2 ) D e v e t − μ a e v e t ] } .
p Fluo ( t | t ′ ) = 1 τ exp ( − t − t ′ τ ) Θ ( t − t ′ ) ,
q e ( r ′ , t ) = ∫ 0 t p Fluo ( t | t ′ ) η e μ a f G ( r s , r ′ , μ a , μ s ′ , n , t ′ ) d t ′ + μ s R G ( r s , r ′ , μ a , μ s ′ , n , t ) ,
Φ e ( r , t ) = ∫ 0 t ∫ V ′ q e ( r ′ , t ′ ) G e ( r ′ , r , μ a e , μ s e ′ , n e , t − t ′ ) d r ′ d t ′ = = Φ e F l u o ( r , t ) + Φ eRaman ( r , t ) ,
Φ e F l u o ( r , t ) = 2 3 μ a f η e v v e τ L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K m y ) sin [ K n ( z + 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e − D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e − μ a v ) ] − 1 × { exp [ − ( K l 2 + K m 2 + K n 2 ) D v t − μ a v t ] − exp [ − ( K l 2 + K m 2 + K n 2 ) D e v e t − μ a e v e t ] } × [ − ( K l 2 + K m 2 + K n 2 ) D v − μ a v + 1 / τ ] − 1 − 2 3 μ a f η e v v e τ L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K m y ) sin [ K n ( z + 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( K l 2 + K m 2 + K n 2 ) D e v e + μ a e v e − 1 / τ ] − 1 × { exp [ − t / τ ] − exp [ − ( K l 2 + K m 2 + K n 2 ) D e v e t − μ a e v e t ] } × [ − ( K l 2 + K m 2 + K n 2 ) D v − μ a v + 1 / τ ] − 1 .
q eFluo ( r ′ , t ′ ) = μ a f τ η e ∫ 0 t ′ exp [ − ( t ′ − t ″ ) τ ] G ( r s , r ′ , μ a , μ s ′ , n , t ″ ) d t ″ = = 2 3 η e μ a f v τ L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x s ) cos ( K l x ′ ) cos ( K m y s ) cos ( K m y ′ ) sin [ K n ( z s + 2 A D ) ] × sin [ K n ( z ′ + 2 A D ) ] [ − ( K l 2 + K m 2 + K n 2 ) D v − μ a v + 1 / τ ] − 1 × { exp [ − ( K l 2 + K m 2 + K n 2 ) D v t ′ − μ a v t ′ ] − exp ( − t ′ / τ ) } ,
R eEBPC ( x , y , λ e , t ) = Φ eFluo ( x , y , z = 0 , λ e , t ) 2 A e + Φ eRaman ( x , y , z = 0 , λ e , t ) 2 A e ,
R eFick ( x , y , t ) = D e ∂ Φ eFluo ( x , y , z = 0 , λ e , t ) ∂ z + D e ∂ Φ eRaman ( x , y , z = 0 , λ e , t ) ∂ z = = R eFluoFick ( x , y , t ) + R eRamanFick ( x , y , t ) ,
R eFluoFick ( x , y , t ) = 2 3 D e μ a f η e v v e τ L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K m y ) K n cos [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e − D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e − μ a v ) ] − 1 × { exp [ − ( K l 2 + K m 2 + K n 2 ) D v t − μ a v t ] − exp [ − ( K l 2 + K m 2 + K n 2 ) D e v e t − μ a e v e t ] } × [ − ( K l 2 + K m 2 + K n 2 ) D v − μ a v + 1 / τ ] − 1 − 2 3 D e μ a f η e v v e τ L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K m y ) K n cos [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( K l 2 + K m 2 + K n 2 ) D e v e + μ a e v e − 1 / τ ] − 1 × { exp [ − t / τ ] − exp [ − ( K l 2 + K m 2 + K n 2 ) D e v e t − μ a e v e t ] } × [ − ( K l 2 + K m 2 + K n 2 ) D v − μ a v + 1 / τ ] − 1 .
Φ e 0 Fluo ( r , t ) = μ a f η e ∫ 0 t ∫ V ′ G ( r s , r ′ , μ a , μ s ′ , n , t ′ ) G e ( r ′ , r , μ a e , μ s e ′ , n e , t − t ′ ) d r ′ d t ′ = − 2 3 μ a f η e v v e L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K m y ) sin [ K n ( z + 2 A e D e ) ] sin [ K n ( z s + 2 A D ) ] × [ ( D e v e − D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e − μ a v ) ] − 1 × { exp [ − ( K l 2 + K m 2 + K n 2 ) D     v t − μ a v t ] − exp ( − ( K l 2 + K m 2 + K n 2 ) D e v e t − μ a e v e t ) } .
Φ eFluo ( r , t ) = [ Φ e 0 Fluo ( r , t ) ] * [ 1 τ exp ( − t / τ ) ] .
R eFluo ( r , t ) = [ R e 0 Fluo ( r , t ) ] * [ 1 τ exp ( − t / τ ) ] .
R e 0 FluoFick ( x , y , t ) = 2 3 D e μ a f η e v v e L x ′ L y ′ L z ′ ∑ l , m , n = 1 ∞ cos ( K l x ) cos ( K m y ) K n cos [ K n ( 2 A e D e ) ] × sin [ K n ( z s + 2 A D ) ] [ ( D e v e − D v ) ( K l 2 + K m 2 + K n 2 ) + ( μ a e v e − μ a v ) ] − 1 × { exp [ − ( K l 2 + K m 2 + K n 2 ) D v t − μ a v t ] − exp [ − ( K l 2 + K m 2 + K n 2 ) D e v e t − μ a e v e t ] } .
μ e λ = μ a b + μ s b + μ s R + μ a f ,
w 1 = ∫ 0 ℓ μ e λ exp ( − μ e λ ℓ ) d ℓ .
w 4 = ∫ 0 t p Fluo ( t ′ | 0 ) d t ′ .
μ e λ e = μ a b e + μ s b e .
ε ( t ) = 1 − R eHeur ( ρ , t ) R eRamanFick ( ρ , t ) .
G ( r , r ′ , t ) = 2 v π a ′ 2 s ′ ∑ l = 1 ∞ ∑ n = 1 ∞ J 0 ( ρ λ l ) J 0 ( ρ ′ λ l ) J 1 2 ( a ′ λ l ) sin [ K n ( z + 2 A D ) ] sin [ K n ( z ′ + 2 A D ) ] exp [ − λ l 2 D v ( t − t ′ ) ] exp [ − K n 2 D v ( t − t ′ ) ] exp [ − μ a v ( t − t ′ ) ] ,
Φ eRamanCyl ( r , t ) = 2 μ s R v v e π a ′ 2 s ′ ∑ l = 1 ∞ ∑ n = 1 ∞ J 0 ( ρ λ l ) J 1 2 ( a ′ λ l ) sin [ K n ( z + 2 A e D e ) ] sin [ K n ( z s + 2 A D ) ] × [ ( D e v e − D v ) ( λ l 2 + K n 2 ) + ( μ a e v e − μ a v ) ] − 1 × { exp [ − ( λ l 2 + K n 2 ) D v t − μ a v t ] − exp [ − ( λ l 2 + K n 2 ) D e v e t − μ a e v e t ] } ,
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