Table 1.
Glossary of Notation of Symbols in the Equations Used in This Work
Symbol Description (x m μ a x , a m ( r ) Absorption coefficient μ s x , s m ′ ( r ) Reduced scattering coefficient r Spatial position κ x , m ( r ) Diffusion coefficient 1 3 [ μ a x , a m ( r ) + 3 [ μ s x , s m ′ ( r ) ] c Velocity of light γ x , m 2 c κ x , m ( r ) Φ x , m ( r , t ) Photon density k Spatial frequency k Magnitude of k q 0 ( r , t ) Q 0 ( k , ω ) Isotropic continuous-wave source term (excitation) q fl ( r , t ) Q fl ( k , ω ) Isotropic continuous-wave source term (emission) τ Fluorescence lifetime of the fluorophore η Fluorescence quantum yield (ratio of photons emitted to photons absorbed) μ a f Absorption extinction coefficient N ( r ) Concentration of fluorophore n ( r ) η μ a f N ( r ) ZBC Zero boundary condition EBC Extrapolated boundary condition
Table 2.
Green’s Function Solution in the Time-Domain Case for Planar Type Geometries, where p = x m
Geometry g geo Γ p Time Domain )Infinite d 2 ( 4 π γ p 2 ) 3 ( t − t ′ ) 5 e − ( μ a p c ( t − t ′ ) + d 2 4 γ p 2 ( t − t ′ ) ) (g01) Semi-infinite half-space 1 2 ( 4 π γ p 2 ) 3 ( t − t ′ ) 5 [ d 1 e − ( μ a p c ( t − t ′ ) + ρ 1 2 4 γ p 2 ( t − t ′ ) ) + d 2 e − ( μ a p c ( t − t ′ ) + ρ 2 2 4 γ p 2 ( t − t ′ ) ) ] (g02) ZBC d 1 = d 2 = z 0 ρ 1 = ρ 2 = ρ EBC d 1 = z 0 d 2 = ( z 0 + 2 z e ) ρ 1 = ρ ρ 2 = ρ̀ Infinite slab (at z = d − e − μ a p c ( t − t ′ ) ( 4 π γ p 2 ) 3 ( t − t ′ ) 5 ∑ n = 0 ∞ [ d 1 e − ρ 1 2 4 γ p 2 ( t − t ′ ) − d 2 e − ρ 2 2 4 γ p 2 ( t − t ′ ) ] (g03) ZBC d 1 = z + n d 2 = z − n ρ 1 = ρ + n ρ 2 = ρ − n EBC d 1 = z̀ + n d 2 = z̀ − n ρ 1 = ρ̀ + n ρ 2 = ρ̀ − n Infinite slab (at z = 0 e − μ a p c ( t − t ′ ) ( 4 π γ p 2 ) 3 ( t − t ′ ) 5 { z 0 e − ρ 2 4 γ p 2 ( t − t ′ ) + ∑ n = 1 ∞ [ d 1 e − ρ 1 2 4 γ p 2 ( t − t ′ ) − d 2 e − ρ 2 2 4 γ p 2 ( t − t ′ ) ] } (g04) ZBC d 1 = z + n ′ d 2 = z − n ′ ρ 1 = ρ + n ′ ρ 2 = ρ − n ′ EBC d 1 = z̀ + n ′ d 2 = z̀ − n ′ ρ 1 = ρ̀ + n ′ ρ 2 = ρ̀ − n ′
Table 3.
Green’s Function Solution in the Time-Domain Case for Circular Type Geometries, where p = x m
Geometry g geo Γ p Time Domain )2D circle, radius a γ p 2 e − μ a p c ( t − t ′ ) π q 2 ∑ n = − ∞ ∞ [ cos ( n θ ) ∑ β n e − γ p 2 β n 2 ( t − t ′ ) β n f n ( β n r ′ , β n q ) ] (g05) Finite cylinder, radius a l 2 γ p 2 e − μ a p c ( t − t ′ ) π q 2 l ∑ k = 1 , odd ∞ e − γ p 2 k 2 π 2 ( t − t ′ ) l 2 ∑ n = − ∞ ∞ [ cos ( n θ ) ∑ β n e − γ p 2 β n 2 ( t − t ′ ) β n f n ( β n r ′ , β n q ) ] (g06) Infinite cylinder, radius a z = z ′ γ p e − μ a p c ( t − t ′ ) 2 π q 2 π ( t − t ′ ) ∑ n = − ∞ ∞ cos ( n θ ) ∑ β n e − γ p 2 β n 2 ( t − t ′ ) β n f n ( β n r ′ , β n q ) (g07) Sphere, radius a γ p 2 e − μ a p c ( t − t ′ ) 2 π q 2 a r ′ ∑ n = 0 ∞ ∑ β n + 1 2 e − γ p 2 β n + 1 2 2 ( t − t ′ ) β n + 1 2 f n + 1 2 ( β n + 1 2 r ′ , β n + 1 2 q ) ( 2 n + 1 ) P n ( cos θ ) (g08) Note: For all the circular geometries, such as the 2D circle, cylinder, and sphere, we have ZBC q = a f n ( β n r ′ , β n q ) = J n ( β n r ′ ) J n + 1 ( β n q ) EBC q = b f n ( β n r ′ , β n q ) = J n ′ ( β n a ) J n ( β n r ′ ) ( J n + 1 ( β n q ) ) 2
Table 4.
Green’s Function Solution in the Frequency-Domain Case for Planar Type Geometries, where p = x m
Geometry G geo Γ p Frequency Domain )Infinite e − j ω t ′ ( 1 + α p d ) e − α p d 2 ( 2 π ) 3 d 2 (G01) Semi-infinite half-space e − j ω t ′ 2 ( 2 π ) 3 / 2 [ ( 1 + α p ρ 1 ) d 1 e − α p ρ 1 ρ 1 3 + ( 1 + α p ρ 2 ) d 2 e − α p ρ 2 ρ 2 3 ] (G02) ZBC d 1 = d 2 = z 0 ρ 1 = ρ 2 = ρ EBC d 1 = z 0 d 2 = ( z 0 + 2 z e ) ρ 1 = ρ ρ 2 = ρ̀ Infinite slab (at z = d − e − j ω t ′ ( 2 π ) 3 ( ∑ n = 0 ∞ [ ( 1 + α p ρ 1 ) d 1 ρ 1 3 e − α p ρ 1 − ( 1 + α p ρ 2 ) d 2 ρ 2 3 e − α p ρ 2 ] ) (G03) ZBC d 1 = z + n d 2 = z − n ρ 1 = ρ + n ρ 2 = ρ − n EBC d 1 = z̀ + n d 2 = z̀ − n ρ 1 = ρ̀ + n ρ 2 = ρ̀ − n Infinite slab (at z = 0 e − j ω t ′ ( 2 π ) 3 ( ( 1 + α p ρ ) z 0 ρ 3 e − α p ρ + ∑ n = 1 ∞ [ ( 1 + α p ρ 1 ) d 1 ρ 1 3 e − α p ρ 1 − ( 1 + α p ρ 2 ) d 2 ρ 2 3 e − α p ρ 2 ] ) (G04) ZBC d 1 = z + n ′ d 2 = z − n ′ ρ 1 = ρ + n ′ ρ 2 = ρ − n ′ EBC d 1 = z̀ + n ′ d 2 = z̀ − n ′ ρ 1 = ρ̀ + n ′ ρ 2 = ρ̀ − n ′
Table 5.
Green’s Function Solution in the Frequency-Domain Case for Circular Type Geometries, where p = x m
Geometry G geo Γ p Frequency Domain )2D circle, radius a e − j ω t ′ ( 2 π ) 3 ∑ n = − ∞ ∞ cos ( n θ ) f n ( α p r ′ ) (G05) Finite cylinder, radius a l e − j ω t ′ π 2 π l ∑ k = 1 , odd ∞ ∑ n = − ∞ ∞ cos ( n θ ) f n ( α p k r ′ ) (G06) Infinite cylinder, radius a z = z ′ e − j ω t ′ ( 2 π ) 3 ∑ n = − ∞ ∞ cos ( n θ ) ∑ β n 1 α p 2 + β n 2 β n g n ( β n r ′ ) (G07) Sphere, radius a e − j ω t ′ 2 ( 2 π ) 3 q r ′ ∑ n = 0 ∞ ( 2 n + 1 ) P n ( cos θ ) f n + 1 / 2 ( r ′ α p ) (G08) Note: For all the circular geometries, such as the 2D circle, cylinder, and sphere, we have ZBC q = a f n ( ε r ′ ) = 1 a I n ( r ′ ε ) I n ( a ε ) g n ( ε r ′ ) = 1 a 2 J n ( r ′ ε ) J n + 1 ( a ε ) EBC q = b f n ( ε r ′ ) = ε I n ( r ′ ε ) I n ( b ε ) F n ′ ( a ε , b ε ) g n ( ε r ′ ) = 1 b 2 J n ′ ( a ε ) J n ( r ′ ε ) ( J n + 1 ( b ε ) ) 2 F n ′ ( a ε , b ε ) = − n a ε F n ( a ε , b ε ) − ( I n − 1 ( a ε ) K n ( b ε ) + K n − 1 ( a ε ) I n ( b ε ) ) F n ( a ε , b ε ) = K n ( a ε ) I n ( b ε ) − I n ( a ε ) K n ( b ε )
Table 6.
Closed-form Expressions for Planar Type Geometries for Integrated Intensity E geo fl ( ξ ) = ( n ζ 2 γ m 2 / ( γ m 2 − γ x 2 ) ) W geo fl ( ξ )
Geometry W geo fl ( ξ ) Infinite ( 1 + σ x d ) e − σ x d − ( 1 + σ m d ) e − σ m d 4 π d 2 (I01) Semi-infinite half-space 1 4 π [ d 1 ( ( 1 + σ x ρ 1 ) e − σ x ρ 1 − ( 1 + σ m ρ 1 ) e − σ m ρ 1 ) ρ 1 3 + d 2 ( ( 1 + σ x ρ 2 ) e − σ x ρ 2 − ( 1 + σ m ρ 2 ) e − σ m ρ 2 ) ρ 2 3 ] (I02) ZBC d 1 = d 2 = z 0 ρ 1 = ρ 2 = ρ EBC d 1 = z 0 d 2 = ( z 0 + 2 z e ) ρ 1 = ρ ρ 2 = ρ̀ Infinite slab 0 < z < d z = d − 1 2 π [ ∑ n = 0 ∞ { d 1 ( ( 1 + σ x ρ 1 ) e − σ x ρ 1 − ( 1 + σ m ρ 1 ) e − σ m ρ 1 ) ρ 1 3 } − ∑ n = 0 ∞ { d 2 ( ( 1 + σ x ρ 2 ) e − σ x ρ 2 − ( 1 + σ m ρ 2 ) e − σ m ρ 2 ) ρ 2 3 } ] (I03) ZBC d 1 = z + n d 2 = z − n ρ 1 = ρ + n ρ 2 = ρ − n EBC d 1 = z̀ + n d 2 = z̀ − n ρ 1 = ρ̀ + n ρ 2 = ρ̀ − n Infinite slab (at z = 0 1 2 π z 0 ( ( 1 + σ x ρ ) e − σ x ρ − ( 1 + σ m ρ ) e − σ m ρ ) ρ 3 + 1 2 π ∑ n = 1 ∞ ( d 1 ( ( 1 + σ x ρ 1 ) e − σ x ρ 1 − ( 1 + σ m ρ 1 ) e − σ m ρ 1 ) ρ 1 3 ) − 1 2 π ∑ n = 1 ∞ ( d 2 ( ( 1 + σ x ρ 2 ) e − σ x ρ 2 − ( 1 + σ m ρ 2 ) e − σ m ρ 2 ) ρ 2 3 ) (I04) ZBC d 1 = z + n ′ d 2 = z − n ′ ρ 1 = ρ + n ′ ρ 2 = ρ − n ′ EBC d 1 = z̀ + n ′ d 2 = z̀ − n ′ ρ 1 = ρ̀ + n ′ ρ 2 = ρ̀ − n ′
Table 7.
Closed-form Expressions for Circular Type Geometries for Integrated Intensity E geo fl ( ξ ) = ( n ζ 2 γ m 2 / ( γ m 2 − γ x 2 ) ) W geo fl ( ξ )
Geometry W geo fl ( ξ ) 2D circle, radius a 1 2 π ∑ n = − ∞ ∞ cos ( n θ ) ( f n ( σ x r ′ ) − f n ( σ m r ′ ) ) (I05) Finite cylinder, radius a l 1 π l ∑ k = 1 , odd ∞ ∑ n = − ∞ ∞ cos ( n θ ) ( f n ( σ x k r ′ ) − f n ( σ m k r ′ ) ) (I06) Infinite cylinder, radius a z = z ′ 1 2 π ∑ n = − ∞ ∞ cos ( n θ ) ∑ β n g n ( β n r ′ ) ( 1 σ x 2 + β n 2 − 1 σ m 2 + β n 2 ) (I07) Sphere, radius a 1 4 π q r ′ ∑ n = 0 ∞ ( f n + 1 2 ( σ x r ′ ) − f n + 1 2 ( σ m r ′ ) ) ( 2 n + 1 ) P n ( cos θ ) (I08) Note: For all circular type geometries, such as the circle, cylinder, and sphere ZBC q = a f n ( ε r ′ ) = 1 a I n ( r ′ ε ) I n ( a ε ) g n ( ε r ′ ) = 1 a 2 J n ( r ′ ε ) J n + 1 ( a ε ) EBC q = b f n ( ε r ′ ) = ε I n ( r ′ ε ) I n ( b ε ) F n ′ ( a ε , b ε ) g n ( ε r ′ ) = 1 b 2 J n ′ ( a ε ) J n ( r ′ ε ) ( J n + 1 ( b ε ) ) 2 F n ′ ( a ε , b ε ) = − n a ε F n ( a ε , b ε ) − ( I n − 1 ( a ε ) K n ( b ε ) + K n − 1 ( a ε ) I n ( b ε ) ) F n ( a ε , b ε ) = K n ( a ε ) I n ( b ε ) − I n ( a ε ) K n ( b ε )
Table 8.
Closed-form Expressions for Planar Type Geometries for Mean Time of Flight 〈 t geo fl 〉 ( ξ ) = 〈 a geo fl 〉 ( ξ ) + ( τ + ζ 2 )
Geometry 〈 a geo fl 〉 ( ξ ) Infinite 1 2 d 2 ( e − σ x d γ x 2 − e − σ m d γ m 2 ) ( ( 1 + σ x d ) e − σ x d − ( 1 + σ m d ) e − σ m d ) (t01) Semi-infinite half-space 1 2 d 1 ρ 1 ( e − σ x ρ 1 υ x − e − σ m ρ 1 υ m ) + d 2 ρ 2 ( e − σ x ρ 2 υ x − e − σ m ρ 2 υ m ) [ d 1 ρ 1 3 ( ( 1 + σ x ρ 1 ) e − σ x ρ 1 − ( 1 + σ m ρ 1 ) e − σ m ρ 1 ) + d 2 ρ 2 3 ( ( 1 + σ x ρ 2 ) e − σ x ρ 2 − ( 1 + σ m ρ 2 ) e − σ m ρ 2 ) ] (t02) ZBC d 1 = d 2 = z 0 ρ 1 = ρ 2 = ρ EBC d 1 = z 0 d 2 = ( z 0 + 2 z e ) ρ 1 = ρ ρ 2 = ρ̀ Infinite slab 0 < z < d z = d 1 2 [ ∑ n = 0 ∞ { d 1 ρ 1 ( e − σ x ρ 1 υ x − e − σ m ρ 1 υ m ) } − ∑ n = 0 ∞ { d 2 ρ 2 ( e − σ x ρ 2 υ x − e − σ m ρ 2 υ m ) } ] [ ∑ n = 0 ∞ { d 1 ρ 1 3 ( ( 1 + σ x ρ 1 ) e − σ x ρ 1 − ( 1 + σ m ρ 1 ) e − σ m ρ 1 ) } − ∑ n = 0 ∞ { d 2 ρ 2 3 ( ( 1 + σ x ρ 2 ) e − σ x ρ 2 − ( 1 + σ m ρ 2 ) e − σ m ρ 2 ) } ] (t03) ZBC d 1 = z + n d 2 = z − n ρ 1 = ρ + n ρ 2 = ρ − n EBC d 1 = z̀ + n d 2 = z̀ − n ρ 1 = ρ̀ + n ρ 2 = ρ̀ − n Note: t ′ t ′ ≠ 0 t ′ 〈 t fl 〉 ( ξ )
Table 9.
Closed-form Expressions for Various Geometries for Mean Time of Flight 〈 t geo fl 〉 ( ξ ) = 〈 a geo fl ( ξ ) 〉 + ( τ + ζ 2 )
Geometry 〈 a geo fl 〉 ( ξ ) Infinite slab (at z = 0 1 2 N D N = z 0 ρ ( e − σ x ρ υ x − e − σ m ρ υ m ) + [ ∑ n = 0 ∞ { d 1 ρ 1 ( e − σ x ρ 1 υ x − e − σ m ρ 1 υ m ) } − ∑ n = 0 ∞ { d 2 ρ 2 ( e − σ x ρ 2 υ x − e − σ m ρ 2 υ m ) } ] D = z 0 ρ 3 ( ( 1 + σ x ρ ) e − σ x ρ − ( 1 + σ m ρ ) e − σ m ρ ) + ∑ n = 1 ∞ d 1 ρ 1 3 ( ( 1 + σ x ρ 1 ) e − σ x ρ 1 − ( 1 + σ m ρ 1 ) e − σ m ρ 1 ) − ∑ n = 1 ∞ d 2 ρ 2 3 ( ( 1 + σ x ρ 2 ) e − σ x ρ 2 − ( 1 + σ m ρ 2 ) e − σ m ρ 2 ) (t04) ZBC d 1 = z + n ′ d 2 = z − n ′ ρ 1 = ρ + n ′ ρ 2 = ρ − n ′ EBC d 1 = z̀ + n ′ d 2 = z̀ − n ′ ρ 1 = ρ̀ + n ′ ρ 2 = ρ̀ − n ′ 2D circle, radius a 1 2 ∑ n = − ∞ ∞ cos ( n θ ) ( 1 υ x f n ′ ( σ x r ′ ) − 1 υ m f n ′ ( σ m r ′ ) ) ∑ n = − ∞ ∞ cos ( n θ ) ( f n ( σ x r ′ ) − f n ( σ m r ′ ) ) (t05) Finite cylinder, radius a l ∑ k = 1 , odd ∞ ∑ n = − ∞ ∞ cos ( n θ ) ( 1 υ x k f n ′ ( σ x k r ′ ) − 1 υ m k f n ′ ( σ m k r ′ ) ) ∑ k = 1 , odd ∞ ∑ n = − ∞ ∞ cos ( n θ ) ( f n ( σ x k r ′ ) − f n ( σ m k r ′ ) ) (t06) Infinite cylinder, radius a z = z ′ 1 2 ∑ n = − ∞ ∞ cos ( n θ ) ∑ β n g n ( β n r ′ ) ( 1 γ x 2 ( σ x 2 + β n 2 ) 3 − 1 γ m 2 ( σ m 2 + β n 2 ) 3 ) ∑ n = − ∞ ∞ cos ( n θ ) ∑ β n g n ( β n r ′ ) ( 1 σ x 2 + β n 2 − 1 σ m 2 + β n 2 ) (t07) Note: t ′ t ′ ≠ 0 t ′ 〈 t fl 〉 ( ξ )
Table 10.
Closed-form Expressions for Circular Type Geometries for Mean Time of Flight 〈 t geo fl 〉 ( ξ ) = 〈 a geo fl 〉 ( ξ ) + ( τ + ζ 2 )
Geometry 〈 a geo fl 〉 ( ξ ) Sphere, radius a 1 2 ∑ n = 0 ∞ ( 1 υ x f n + 1 2 ′ ( σ x r ′ ) − 1 υ m f n + 1 2 ′ ( σ m r ′ ) ) ( 2 n + 1 ) P n ( cos θ ) ∑ n = 0 ∞ ( f n + 1 2 ( σ x r ′ ) − f n + 1 2 ( σ m r ′ ) ) ( 2 n + 1 ) P n ( cos θ ) (t08) Note: t ′ t ′ ≠ 0 t ′ 〈 t fl 〉 ( ξ ) Note: For all circular type geometries, such as the circle, cylinder, and sphere ZBC q = a f n ( ε r ′ ) = 1 a I n ( r ′ ε ) I n ( a ε ) g n ( ε r ′ ) = 1 a 2 J n ( r ′ ε ) J n + 1 ( a ε ) f n ′ ( r ′ ε ) = 1 a U n ( ε r ′ , a ) EBC q = b f n ( ε r ′ ) = ε I n ( r ′ ε ) I n ( b ε ) F n ′ ( a ε , b ε ) g n ( ε r ′ ) = 1 b 2 J n ′ ( a ε ) J n ( r ′ ε ) ( J n + 1 ( b ε ) ) 2 f n ′ ( r ′ ε ) = ( I n ( r ′ ε ) I n ( b ε ) + ε U n ( ε r ′ , b ) ) F n ′ ( a ε , b ε ) + ε I n ( r ′ ε ) I n ( b ε ) V n ( ε a , ε b ) U n ( r ′ ε , q ) = r ′ I n ( q ε ) I n − 1 ′ ( r ′ ε ) − q I n − 1 ′ ( q ε ) I n ( r ′ ε ) ( I n ( q ε ) ) 2 F n ′ ( a ε , b ε ) = − n a ϵ F n ( a ε , b ε ) − ( I n − 1 ( a ε ) K n ( b ε ) + K n − 1 ( a ε ) I n ( b ε ) ) V n ( a ε , b ε ) = ( n ( 2 n + 1 ) a ε 2 − a ) F n ( a ε , b ε ) + b F n − 1 ( a ε , b ε ) + n − 1 ε ( I n − 1 ( a ε ) K n ( b ε ) + K n − 1 ( a ε ) I n ( b ε ) ) − b n a ε ( I n ( a ε ) K n − 1 ( b ε ) + K n ( a ε ) I n − 1 ( b ε ) ) F n ( a ε , b ε ) = K n ( a ε ) I n ( b ε ) − I n ( a ε ) K n ( b ε )
Table 11.
Glossary of Notation of Symbols Used in Tables 2 10
Symbol Formula Symbol Formula ξ = x 2 + y 2 σ x , m = μ a x , a m c γ x , m ρ = ξ 2 + z 0 2 α x , m = μ a x , a m c + j ω γ x , m z + n = ( 2 n + 1 ) d + z 0 α x k , m k = ( α x , m 2 + k 2 π 2 l 2 ) z − n = ( 2 n + 1 ) d − z 0 r ′ Radial position of the source. z + n ′ = 2 n d + z 0 β j Positive root of J j ( β j a ) = 0 j = n n + 1 2 z − n ′ = 2 n d − z 0 ζ 2 = γ m 2 − γ x 2 c ( γ m 2 μ a x − γ x 2 μ a m ) z̀ + n = ( 2 n + 1 ) ( d + 2 z e ) + z 0 z̀ − n = ( 2 n + 1 ) ( d + 2 z e ) − z 0 z̀ + n ′ = 2 n ( d + 2 z e ) + z 0 z̀ − n ′ = 2 n ( d + 2 z e ) − z 0
Table 12.
Glossary of Notation of Symbols Used in Tables 2 10
Symbol Formula Symbol Formula ρ + n = ξ 2 + z + n 2 υ x , m = γ x , m μ a x , a m c ρ − n = ξ 2 + z − n 2 γ x , m = c 3 ( μ a x , a m + μ s x , s m ′ ) ρ + n ′ = ξ 2 + z + n ′ 2 σ x k , m k = ( μ a x , a m c γ x , m 2 + k 2 π 2 l 2 ) ρ − n ′ = ξ 2 + z − n ′ 2 υ x k , m k = ( μ a x , a m c + γ x , m 2 k 2 π 2 l 2 ) Bessel function — J ν ( x ) = ∑ r = 0 ∞ ( − 1 ) r ( 1 2 x ) ν + 2 r r ! Γ ( ν + r + 1 ) Modified Bessel function — I ν ( x ) = ∑ r = 0 ∞ ( 1 2 x ) ν + 2 r r ! Γ ( ν + r + 1 ) Legendre polynomial — P n ( x ) = 1 2 n n ! d n d x n [ ( x 2 − 1 ) n ]
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