Optical Fiber Sensors for Biomedical Applications

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680 L.C.L. Chin et al.described by an escape function, E, which is equivalent to the photon current acrossthe boundary at z = 0, is given by Fick’s Law as:(−D d )dz φ (17.16)Combining Eqs. (17.15) into (17.16) gives the escape function E(κ, θ, z, r):E (κ, θ, z, r) = 1 [ z exp (−μeff k)(μ4π k 2 eff + 1 )− z p exp (μ eff l)(μk l 2 eff + 1 )]l(17.17)where k and l are the positive roots of:z = 0k 2 = (r − κ cos θ) 2 + κ 2 sin 2 θ + z 2(17.18a)l 2 = (r − κ cos θ) 2 + κ 2 sin 2 θ + z 2 p(17.18b)Using Eqs. (17.15) and (17.17), the change in reflectance, R(r), due to a smallabsorbing perturbation, μ a in a small incremental volume, dV, is then:R (r) = μ a φ ( r ′) dV E ( r ′ , r ) (17.19)The mean photon-visit depth, 〈z〉 r , for a source-detector separation of r is thengiven by:∫V〈z〉 r =φ ( r ′) E ( r ′ , r ) zdV∫V φ (r′ ) dV E (r ′ (17.20), r) dVUsing Eq. √ (17.20), Patterson et al. [12] examined the change in 〈z〉 r as a functionof δ 1 /2 = 1 / μ eff (Fig. 17.10). They demonstrated that, using a simple empiricalrelationship, a reasonable estimation of 〈z〉 r could be determined from knowledgeof r and δ, alone:〈z〉 r = (rδ)1 /22(17.21)From Fig. 17.10 it is clear that short source-detector separations (lower line)result in shorter penetration depths while long source-detector separations (upperline) result in deeper sampling volumes. Furthermore, the mean penetration depthincreases with decreasing optical attenuation.Multi-distance reflectance probes that rely on diffusion theory for data fitting typicallyinvolved source-detector separations of up to ∼2 cm. Such a large probe sizemay pose serious problems when the optical properties vary significantly over theprobe sampling volume (e.g., when evaluating layered structures such as epithelialtissue, or sampling heterogeneous tumors with small localized regions of hypoxia).
17 Optical Fiber Sensors for Biomedical Applications 681Fig. 17.10 Plot of 〈z〉 r ,themean photon visit depth vs.δ 1/2 , the square root of thepenetration depth. Thesymbols representexperimental measurements.The lines are a linear fit of theresulting data for eachsource-detector separation,demonstrating an empiricalrelationship of 〈z〉 r = (rδ)1/22as per Eq. (17.21). (Reprintedwith permission from [12])Mean depth (mm)765432r = 20 mmr = 10 mmr = 5 mm100 0.5 1.0 1.5 2.0 2.5 3.0δ 1/2As demonstrated in Section 17.4.2.1, sensors with shorter source-detector separations(2–3 mm) must be employed to confine the volume of interrogation. Anexample of one such sensor (employed by Finlay and Foster [14]) is shown inFig. 17.11 and was used to reveal significant heterogeneities in oxygen saturationin small murine tumor models. However, in these cases diffusion theory isno longer valid. As such, when short-detector spacing is required, alternative forwardmodels and inversion algorithms must be employed during fitting for accuraterecovery of optical properties and their spectral dependencies. A brief summary ofthe different modeling approaches employed for short s–d distances is presented inTable 17.2.While an exhaustive explanation of all models is beyond the scope of this chapter,in general, the approaches can be divided into improved analytical solutions of theradiative transport equation such as the δ-P1 approximation, Born approximationand P3 approximations, or fast inversion algorithms based on Monte Carlo generatedDDDDFig. 17.11 Schematic of asmall SD separationreflectance sensor. The source(S) and detector (D)separation distances (mm) areindicated in the figure (afterFinlay and Foster[14])DSD
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