Diffuse photon remission associated with the center-illuminated-area-detection geometry. III. Perspectives on the patterns of saturation.

Understanding scattering insensitiveness in diffuse reflectance spectroscopy (DRS) will be useful to enhancing the spectral specificity to absorption. In DRS based on center-illuminated-area-detection (CIAD), the scattering response can saturate as the relative strength of scattering with respect to the collection size, represented by a dimensionless reduced scattering, increases over a threshold. However, the formation of saturation versus the same range of dimensionless reduced scattering may differ between a fixed reduced scattering over an increasing collection size (case 1) and an increasing reduced scattering over a fixed collection size (case 2), due to the absorption. Part III demonstrates the differences of the scattering saturation as well as the effect of absorption on it in the CIAD geometry between the two cases while assessed over the same range of the dimensionless reduced scattering. A model allows predicting the absorption-dependent levels of saturation and the corner parameters of saturation transition. When assessed for the absorption coefficient to vary over [0.001,0.01,0.1,1]m m -1, the model-predicted levels of saturation agree with MC results with ≤2.2% error in both cases. In comparison, the model-predicted corner parameters of saturation show much different agreement with MC results in the two cases, suggesting that the saturation pattern is much better formed in one than in the other. Experiments conforming to the CIAD geometry support the discrepancy of the saturating patterns between the two cases.

Diffuse photon remission associated with the center-illuminated-area-detection geometry. II. Approach to the time-domain model.

This part proposes a model of time-dependent diffuse photon remission for the center-illuminated-area-detection (CIAD) geometry, by virtue of area integration of the radially resolved time-dependent diffuse photon remission formulated with the master-slave dual-source scheme demonstrated in Part I for steady-state measurements. The time-domain model is assessed against Monte Carlo (MC) simulations limiting to only the Heyney-Greenstein scattering phase function for CIAD of physical scales and medium properties relevant to single-fiber reflectance (SfR) and over a 2 ns duration, in compliance with the timespan of the only experimental report of SfR demonstrated with a 50 µm gradient index fiber. The time-domain model-MC assessments are carried out for an absorption coefficient ranging three orders of magnitude over [0.001,0.01,0.1,1]m m -1 at a fixed scattering, and a reduced scattering coefficient ranging three orders of magnitude over [0.01,0.1,1,10]m m -1 at a fixed absorption, among others. Photons of shorter and longer propagation times, relative to the diameter of the area of collection, respond differently to the scattering and absorption changes. Limited comparisons of MC between CIAD and a top-hat geometry as the idealization of SfR reveal that the time-domain photon remissions of the two geometries differ appreciably in only the early arriving photons.

Diffuse photon remission associated with the center-illuminated-area-detection geometry: Part I, an approach to the steady-state model.

Diffuse photon remission associated with the center-illuminated-area-detection (CIAD) geometry has been useful for non-contact sensing and may inform single-fiber reflectance (SfR). This series of work advances model approaches that help enrich the understanding and applicability of the photon remission by CIAD. The general approach is to derive the diffuse photon remission by the area integration of the radially resolved diffuse reflectance while limiting the analysis to a medium exhibiting only the Heyney-Greenstein (HG) scattering phase function. Part I assesses the steady-state photon remission in CIAD over a reduced scattering scaled diameter of µ s ' d a r e a ∈[0.5×10-3,103] that covers the range extensively modeled for SfR. The corresponding radially resolved diffuse reflectance is obtained by concatenating an empirical expression for the semi-ballistic region near the point-of-illumination and a formula utilizing a master-slave dual-source scheme over the semi-diffusive to a diffusive regime while being constrained by an extrapolated zero-boundary condition. The terminal algebraic photon remission is examined against Monte Carlo simulations for an absorption coefficient over [0.001,1]m m -1, a reduced scattering coefficient over [0.01,1000]m m -1, a HG scattering anisotropy factor within [0.5,0.95], and a diameter of the area of collection ranging [50,1000]µm. The algebraic model is also applied to phantom data acquired over a ∼2c m non-contact CIAD configuration and with a 200 µm SfR probe. The model approach will be extended in a subsequent work towards the time-of-flight characteristics of CIAD.

Diffuse photon remission from thick opaque media of the high absorption/scattering ratio beyond what is accountable by the Kubelka-Munk function.

The Kubelka-Munk (KM) theory of diffuse photon remission from opaque media is widely applied to quality-control processes. Recent works based on radiative transfer revealed that the KM function as the backbone parameter of the method may saturate at strong absorption to cause the KM approach to be unfit to predict the change of diffuse reflectance from the medium at strong absorption. We demonstrate by empirical means based on Monte Carlo results that diffuse photon remission from a strong-absorbing medium depends simply upon the absorption/scattering ratio when evaluated over a large area centered at the point of illumination differing in geometry from those convenient for the KM approach. Our empirical prediction gives ∼11% mean errors of the diffuse photon remission from thick opaque medium having an absorption coefficient ranging 0.001 to up to 1000 times stronger than the reduced-scattering coefficient. A slight modification to the KM function in terms of the absorption weighting and absorption-scattering coupling for use within the KM approach also noticeably improves the prediction of diffuse photon remission from thick opaque medium of strong absorption. Our empirical model and the KM approach using the modified KM function were compared against measurements from a thick opaque medium, of which the absorption coefficient was changed over four orders of magnitude.

Diffuse photon-remission associated with single-fiber geometry may be a simple scaling of that collected over the same area when under centered-illumination.

Robust models for single-fiber reflectance (SFR) are relatively complex [Opt. Lett.45, 2078 (2020)OPLEDP0146-959210.1364/OL.385845] due to overlapping of the illumination and collection areas that entails probability weighting of the spatial integration of photon-remission. We demonstrate, via analytical means for limiting cases and Monte Carlo simulation of broader conditions, that diffuse photon-remission collected by single-fiber geometry may be scaled over the center-illuminated photon-remission. We specify for a medium revealing Henyey-Greenstein (HG) scattering anisotropy that the diffuse photon-remission from a sub-diffusive area of a top-hat illumination is ∼84.9% of that collected over the same area when under a centered-illumination. This ratio remains consistent over a reduced-scattering fiber-size product of µs'dfib=[10-5,100], for absorption varying 3 orders of magnitude. When applied to hemoglobin oxygenation changes induced in an aqueous phantom using a 200 µm single-fiber probe, the center-illumination-scaled model of SFR produced fitting results agreeing with reference measurements.

Laparoscopic diffuse reflectance spectroscopy of an underlying tubular inclusion: a phantom study.

We demonstrate diffuse reflectance spectroscopy (DRS) of a subsurface tubular inclusion by using a fiber probe having a single source-detector pair attached to a laparoscopic bipolar device. A forward model was also developed for DRS sensing of an underlying long absorbing tubular inclusion set in parallel to the tissue surface, normal to the line of sight of the source-detector pair, and equidistant from the source and the detector. The model agreed with measurements performed at 500 nm and using a 10 mm source-detector separation (SDS) on an aqueous tissue phantom embedding a tubing of 2 or 4 mm inner diameter that contained 9.1% to 33.3% red dye at a depth of up to 11.5 mm. When tested on solid phantoms using the 10 mm SDS, a tubular inclusion of $ \ge 3\;{\rm mm}$≥3mm inner diameter containing 0.05% red dye at a background absorption coefficient of $ 0.021\;{\rm mm}^{-1} $0.021mm-1 caused $ \ge 8\% $≥8% change of the signal at 500 nm versus the baseline when the inclusion was shallower than 5 mm. When assessed on avian muscle tissue having a 4 mm tubular inclusion embedded at an edge depth of 2 mm, DRS with the 10 mm SDS differentiated the following contents of the inclusion: 33.3% red dye (mimicking blood), 33.3% green dye, 33.3% yellow dye (mimicking bile), water (mimicking urine), and air.

Simple analytical total diffuse reflectance over a reduced-scattering-pathlength scaled dimension of [10-5, 10-1] from a medium with HG scattering anisotropy.

Model approximation is necessary for reflectance assessment of tissue at sub-diffusive to non-diffusive scale. For tissue probing over a sub-diffusive circular area centered on the point of incidence, we demonstrate simple analytical steady-state total diffuse reflectance from a semi-infinite medium with the Henyey-Greenstein (HG) scattering anisotropy (factor $g$g). Two physical constraints are abided to: (1) the total diffuse reflectance is the integration of the radial diffuse reflectance; (2) the radial and total diffuse reflectance at $g \gt {0}$g>0 analytically must resort to their respective forms corresponding to isotropic scattering as $g$g becomes zero. Steady-state radial diffuse reflectance near the point of incidence from a semi-infinite medium of $g \approx 0$g≈0 is developed based on the radiative transfer for isotropic scattering, then integrated to find the total diffuse reflectance for $g \approx 0$g≈0. The radial diffuse reflectance for $g \ge 0.5$g≥0.5 is semi-empirically formulated by comparing to Monte Carlo simulation results and abiding to the second constraint. Its integration leads to a total diffuse reflectance for $g \ge 0.5$g≥0.5 that is also bounded by the second constraint. Over a collection diameter of the reduced-scattering pathlength ($1/\mu _s^{ \prime}$1/µs') scaled size of [${{10}^{ - 5}}$10-5, ${{10}^{ - 1}}$10-1] for $g = [{0.5},{0.95}]$g=[0.5,0.95] and the absorption coefficient as strong as the reduced scattering coefficient, the simple analytical total diffuse reflectance is found to be accurate, with an average error of 16.1%.

Transcutaneous transmission of photobiomodulation light to the spinal canal of dog as measured from cadaver dogs using a multi-channel intra-spinal probe.

The target level photobiomodulation (PBM) irradiances along the thoracic to lumbar segment of the interior spinal canal in six cadaver dogs resulting from surface illumination at 980 nm were measured. Following a lateral hemi-laminectomy, a flexible probe fabricated on a plastic tubular substrate of 6.325 mm diameter incorporating nine miniature photodetectors was embedded in the thoracic to lumbar segment of the spinal canal. Intra-spinal irradiances at the nine photodetector sites, spanning an approximate 8 cm length caudal to T13, were measured for various applied powers of continuous wave (CW) surface illumination at 980 nm with a maximal power of 10 W corresponding to a surface irradiance of 3.14 W/cm2. The surface illumination conditions differed in skin transmission when the probe was off-contact with tissue and probe-skin contact when the skin was in place. For each condition of surface illumination, the beam was directed to respectively T13 (surface site 1), a spinal column site 4 cm caudal to T13 (surface site 5), and a spinal column site 8 cm caudal to T13 (surface site 9). Off-contact surface irradiation of 3.14 W/cm2 at surface sites 1, 5, and 9 transmitted respectively 234.0 ± 120.7 µW/cm2, 230.7 ± 178.3 µW/cm2, and 130.2 ± 169.6 µW/cm2 to the spinal canal without the skin, and respectively 35.7 ± 33.2 µW/cm2, 50.9 ± 75.3 µW/cm2, and 15.7 ± 16.3 µW/cm2 with the skin. Transmission with skin was as low as 12% of the transmission without the skin. On-contact surface irradiation of 3.14 W/cm2 at surface sites 1, 5, and 9 transmitted respectively 44.6 ± 43.1 µW/cm2, 85.4 ± 139.1 µW/cm2, and 22.0 ± 23.6 µW/cm2 to the spinal canal. On-contact application increased transmission by a maximum of 67% comparing to off-contact application. The information gathered highlights the need to clinically consider the impact of skin transmission and on-contact application technique when attempting to treat spinal cord disease with PBM.

Simple empirical master-slave dual-source configuration within the diffusion approximation enhances modeling of spatially resolved diffuse reflectance at short-path and with low-scattering from a semi-infinite homogeneous medium: erratum.

We correct one typographical error of three equations in Appl. Opt.56, 1447 (2017)APOPAI0003-693510.1364/AO.56.001447.

Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. VI. Time-domain analysis.

Part VI analytically examines time-domain (TD) photon diffusion in a homogeneous medium enclosed by a "concave" circular cylindrical applicator or enclosing a "convex" circular cylindrical applicator, both geometries being infinite in the longitudinal dimension. The aim is to assess characteristics of TD photon diffusion, in response to a spatially and temporally impulsive source, versus the line-of-sight source-detector distance along the azimuthal or longitudinal direction on the concave or convex medium-applicator interface. By comparing to their counterparts evaluated along a straight line on a semi-infinite medium-applicator interface versus the same source-detector distance, the following patterns are indicated: (1) the peak photon fluence rate is always reached sooner in concave and later in convex geometry; (2) the peak photon fluence rate decreases slower along the azimuthal and faster along the longitudinal direction on the concave interface, and conversely on the convex interface; (3) the total photon fluence decreases slower along the azimuthal and faster along the longitudinal direction on the concave interface, and conversely on the convex interface; (4) the ratio between the peak photon fluence rate and the total fluence is always greater in concave geometry and smaller in convex geometry. The total fluence is equivalent to the steady-state photon fluence analyzed in Part I [J. Opt. Soc. Am. A27, 648 (2010)10.1364/JOSAA.27.000648JOAOD61084-7529]. The patterns of peak fluence rate, time to reaching peak fluence rate, and the ratio of these two, correspond to those of AC amplitude, phase, and modulation depth of frequency-domain results demonstrated in Part IV [J. Opt. Soc. Am. A29, 1445 (2012)10.1364/JOSAA.29.001445JOAOD61084-7529].

Percutaneous single-fiber reflectance spectroscopy of canine intervertebral disc: is there a potential for in situ probing of mineral degeneration?

BACKGROUND AND OBJECTIVES: Intervertebral disc herniation is a common disease in chondrodystrophic dogs, and a similar neurologic condition also occurs in humans. Percutaneous laser disc ablation (PLDA) is a minimally invasive procedure used increasingly for prevention of disc herniation. Currently, PLDA is performed on thoracolumbar discs with the same laser energy applied regardless of the differing extent of degeneration among mineralized discs. In a previous study performed on 15 normal and 6 degenerated intervertebral discs in chondrodystrophoid canine species, it was demonstrated that percutaneous single-fiber reflectance spectroscopy (SfRS) detected increased light scattering from mineralized intervertebral discs when comparing to normal discs. The objective of this study is to evaluate how SfRS evaluation of mineralized discs in situ fairs with X-ray radiography and computed tomography (CT) diagnoses and if SfRS sensing of the scattering changes correlates with the level of mineral degeneration in nucleus pulposus. MATERIALS AND METHODS: Percutaneous SfRS was performed on a total of 28 intervertebral discs of three dogs post-mortem, through a 20 gauge spinal needle standard to PLDA. The raw SfRS measurement was normalized to extract a dimension-less spectral intensity profile, from which the average over 600-900 nm was used as the SfRS intensity index to compare among the measured discs. The discs were imaged prior to percutaneous SfRS by radiography and CT, and harvested after percutaneous SfRS for histopathologic examinations. RESULTS: Five among 10 discs of dog #1, six among 9 discs of dog #2, and nine out of 9 discs of dog #3 were determined by histopathology to have central focal or multi-focal areas of mineralization occupying 5-75% of the examined area of nucleus pulposus. The overall numbers of discs with detectable and undetectable central mineralization were 20 and 8, respectively. CT resulted in one false positive (FP) and four false negative (FN) diagnoses for dog #1, three FP and zero FN diagnoses for dog #2, and zero FP and one FN diagnosis for dog #3. Of the total 28 discs the CT had an overall positive predictive value (PPV) of 78.8% and an overall negative predictive value (NPV) of 44.4%. X-ray radiography gave five FN diagnoses for dog #1, two FN diagnoses for dog #2, and eight FN diagnoses for dog #3. Of the total 28 discs the radiography had an overall PPV of 100% and an overall NPV of 30.4%. The receiver-operating-characteristic analysis of the SfRS measurement was performed on 24 discs that had a central mineralization not greater than 50%. An area-under-curve of 0.6758 infers that the SfRS intensity weakly indicates the level of mineralization. CONCLUSIONS: Percutaneous SfRS may be useful as an in situ sensing tool for assessing the level of mineral degeneration in intervertebral discs for the prospect of disc-specific dosage adjustment in PLDA.

A geometric-sensitivity-difference method improves object depth-localization for continuous-wave fluorescence diffuse optical tomography: an in silico study in an axial outward-imaging geometry.

A geometric-sensitivity-difference (GSD) based reconstruction method is demonstrated in fluorescence diffuse optical tomography (FDOT) for improving the depth-localization of objects. The GSD method optimizes the data-model fit based on paired-measurements between source-detector pairs sharing either the source or the detector channel, as comparing to conventional methods that optimize the data-model fit based on un-paired measurements of individual source-detector pairs. This in silico study is limited to continuous-wave and 2-dimension, for a circular-array outward-imaging geometry of which the native sensitivity of measurement varies strongly with respect to the depth of the object. The outcomes of GSD method are compared to that of two conventional methods: one is the baseline method which does not involve any scheme to compensate the variation of native sensitivity; the other applies a depth-adapted weight to counteract the depth-variance of the native sensitivity. These three methods were evaluated using synthetic data corresponding to the following conditions of the object: (1) Single object with a 3-folds of positive contrast of fluorescence over the background was set at edge-depths of 0.5 mm, 5 mm and 10 mm; (2) Two objects with identical 3-folds of positive or 1/3-folds of negative contrast of fluorescence over the background were set at a fixed edge-depth of 10 mm and different azimuthal separations of 45 degree, 135 degree, and 180 degree; (3) Two objects with identical 3-folds of positive or 1/3-folds of negative contrast of fluorescence over the background were set at a fixed azimuthal separation of 90° and at edge-depths of 0.5 mm, 5 mm and 10 mm. The GSD method outperforms the other two methods in localizing a single anomaly and resolving two anomalies, for the anomaly possessing either the 3 folds positive or 1/3-folds negative contrast of fluorescence over the background. The case of objects with negative contrast over the background has specific implications to imaging zinc-specific fluorophore uptake in prostate.

Novel needle-probe single-fiber reflectance spectroscopy to quantify sub-surface myoglobin forms in beef psoas major steaks during retail display.

Meat discoloration starts at the interface between the bright red oxymyoglobin layer and the interior deoxymyoglobin layer. Currently, limited tools are available to characterize myoglobin forms formed within the sub-surface of meat. The objective was to demonstrate a needle-probe based single-fiber reflectance (SfR) spectroscopy approach for characterizing sub-surface myoglobin forms of beef psoas major muscles during retail storage. A 400-µm fiber was placed in a 17-gauge needle, and the assembly was inserted into the muscle at five depths of 1 mm increment and 1 cm lateral shift. Metmyoglobin content increased at all depths during display and content at 1 mm was greater compared to that of 2 to 5 mm depth. The a* values decreased (P < 0.05) during retail display aligning with the sub-surface formation of metmyoglobin. In summary, the results suggest that needle-probe SfR spectroscopy can determine interior myoglobin forms and characterize meat discoloration.

Toward transduodenal diffuse optical tomography of proximal pancreas.

We demonstrate the feasibility of diffuse optical tomography (DOT) of the proximal pancreas by using optical applicator channels deployed longitudinally along the exterior surface of a duodenoscope. As the duodenum that nearly encircles the proximal pancreas forms a natural "C-loop" that is approximately three-quarters of a circle of 5-6 cm in diameter, a multichannel optical applicator attached to a duodenoscope has the potential to perform transduodenal DOT sampling of the bulk proximal pancreas wherein most cancers and many cystic lesions occur. The feasibility of transduodenal DOT is demonstrated on normal porcine pancreas tissues containing an introduced gelatinous inclusion of approximately 3 cm in diameter, by using nine source channels and six detector channels attached to a duodenoscope. Concurrent ultrasonography of the gelatinous inclusion in the porcine pancreas parenchyma provided a coarse, albeit indispensable, anatomic prior to transduodenal DOT in reconstructing a contrast of optical properties in the pancreas.

Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. V. Steady-state fluorescence.

As Part V in our series, this paper examines steady-state fluorescence photon diffusion in a homogenous medium that contains a homogenous distribution of fluorophores, and is enclosed by a "concave" circular cylindrical applicator or is enclosing a "convex" circular cylindrical applicator, both geometries being infinite in the longitudinal dimension. The aim is to predict by analytics and examine with the finite-element method the changing characteristics of the fluorescence-wavelength photon-fluence rate and the ratio (sometimes called the Born ratio) of it versus the excitation-wavelength photon-fluence rate, with respect to the source-detector distance. The analysis is performed for a source and a detector located on the medium-applicator interface and aligned either azimuthally or longitudinally in both concave and convex geometries. When compared to its steady-state counterparts on a semi-infinite medium-applicator interface with the same line-of-sight source-detector distance, the fluorescence-wavelength photon-fluence rate reduces faster along the longitudinal direction and slower along the azimuthal direction in the concave geometry, and conversely in the convex geometry. However, the Born ratio increases slower in both azimuthal and longitudinal directions in the concave geometry and faster in both directions in the convex geometry, respectively, when compared to that in the semi-infinite geometry.

Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. IV. Frequency-domain analysis.

Part IV examines frequency-domain photon diffusion in a homogeneous medium enclosed by a "concave" circular cylindrical applicator or enclosing a "convex" circular cylindrical applicator, both geometries being infinite in the longitudinal dimension. The aim is to assess by analogical and finite-element methods the changes of AC amplitude, modulation depth, and phase with respect to the line-of-sight source-detector distance for a source and a detector located along the azimuthal or longitudinal direction on the concave or convex medium-applicator interface. By comparing to their counterparts along a straight line on a semi-infinite medium-applicator interface, for the same line-of-sight source-detector distance, it is found that: (1) the decay-rate of AC photon fluence is smaller along the azimuthal direction and greater along the longitudinal direction on the concave interface, (2) the decay-rate of AC photon fluence is greater along the azimuthal direction and smaller along the longitudinal direction on the convex interface, (3) the modulation depth along both azimuthal and longitudinal directions decays more slowly on the concave interface and faster on the convex interface, and (4) the phase along both azimuthal and longitudinal directions increases more slowly on the concave interface and faster on the convex interface.

Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. III. Synthetic study of continuous-wave photon fluence rate along unique spiral paths.

This is Part III of the work that examines photon diffusion in a scattering-dominant medium enclosed by a "concave" circular cylindrical applicator or enclosing a "convex" circular cylindrical applicator. In Part II of this work Zhang et al. [J. Opt. Soc. Am. A, 66 (2011)] predicted that, on the tissue-applicator interface of either "concave" or "convex" geometry, there exists a unique set of spiral paths, along which the steady-state photon fluence rate decays at a rate equal to that along a straight line on a planar semi-infinite interface, for the same line-of-sight source-detector distance. This phenomenon of steady-state photon diffusion is referred to as "straight-line-resembling-spiral paths" (abbreviated as "spiral paths"). This Part III study develops analytic approaches to the spiral paths associated with geometry of a large radial dimension and presents spiral paths found numerically for geometry of a small radial dimension. This Part III study also examines whether the spiral paths associated with a homogeneous medium are a good approximation for the medium containing heterogeneity. The heterogeneity is limited to an anomaly that is aligned azimuthally with the spiral paths and has either positive or negative contrast of the absorption or scattering coefficient over the background medium. For a weak-contrast anomaly the perturbation by it to the photon fluence rate along the spiral paths is found by applying a well-established perturbation analysis in cylindrical coordinates. For a strong-contrast anomaly the change by it to the photon fluence rate along the spiral paths is computed using the finite-element method. For the investigated heterogeneous-medium cases the photon fluence rate along the homogeneous-medium associated spiral paths is macroscopically indistinguishable from, and microscopically close to, that along a straight line on a planar semi-infinite interface.

Diffuse photon remission along unique spiral paths on a cylindrical interface is modeled by photon remission along a straight line on a semi-infinite interface.

We demonstrate that, for a long cylindrical applicator that interfaces concavely or convexly with a scattering-dominant medium, a unique set of spiral-shaped directions exist on the tissue-applicator interface, along which the diffuse photon remission is essentially modeled by the photon remission along a straight line on a semi-infinite interface. This interesting phenomenon, which is validated in steady state in this work by finite-element and Monte Carlo methods, may be particularly useful for simplifying deeper-tissue sensing in endoscopic imaging geometry.

Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. II. Quantitative examinations of the steady-state theory.

This is Part II of the work that examines photon diffusion in a homogenous medium enclosed by a concave circular cylindrical applicator or enclosing a convex circular cylindrical applicator. Part I of this work [J. Opt. Soc. Am. A 27, 648 (2010)] analytically examined the steady-state photon diffusion between a source and a detector for two specific cases: (1) the detector is placed only azimuthally with respect to the source, and (2) the detector is placed only longitudinally with respect to the source, in the infinitely long concave and convex applicator geometries. For the first case, it was predicted that the decay rate of photon fluence would become smaller in the concave geometry and greater in the convex geometry than that in the semi-infinite geometry for the same source-detector distance. For the second case, it was projected that the decay rate of photon fluence would be greater in the concave geometry and smaller in the convex geometry than that in the semi-infinite geometry for the same source-detector distance. This Part II of the work quantitatively examines these predictions from Part I through several approaches, including (a) the finite-element method, (b) the Monte Carlo simulation, and (c) experimental measurement. Despite that the quantitative examinations have to be conducted for finite cylinder applicators with large length-to-radius ratio to approximate the infinite-length condition modeled in Part I, the results obtained by these quantitative methods for two concave and three convex applicator dimensions validated the qualitative trend predicted by Part I and verified the quantitative accuracy of the analytic treatment of Part I in the diffusion regime of the measurement, at a given set of absorption and reduced scattering coefficients of the medium.

Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. I. Steady-state theory.

This work presents an analytic treatment for photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. Focusing initially on the steady-state condition, the photon diffusion in these two geometries is solved in cylindrical coordinates by using modified Bessel functions and by applying the extrapolated boundary condition. For large cylinder diameter, the analytic solutions may be simplified to a format employing the physical source and its image source with respect to a semi-infinite geometry and a radius-dependent term to account for the shape and dimension of the cylinder. The analytic solutions and their approximations are evaluated numerically to demonstrate qualitatively the effect of the applicator curvature--either concave or convex--and the radius on the photon fluence rate as a function of the source-detector distance, in comparison with that in the semi-infinite geometry. This work is subjected to quantitative examination in a coming second part and possible extension to time-resolved analysis.