Small Source-detector Separations Research Articles

A refined analytical model for reconstruction problems in diffuse reflectance spectroscopy

A refined analytical model of spatially resolved diffuse reflectance with small source-detector separations (SDSs) for the in vivo skin studies is proposed. Compared to the conventional model developed by Farrell et al., it accounts for the limited acceptance angle of the detector fiber. The refined model is validated in the wide range of optical parameters by Monte Carlo simulations of skin diffuse reflectance at SDSs of units of mm. Cases of uniform dermis and two-layered epidermis-dermis structures are studied. Higher accuracy of the refined model compared to the conventional one is demonstrated in the separate, constraint-free reconstruction of absorption and reduced scattering spectra of uniform dermis from the Monte Carlo simulated data. In the case of epidermis-dermis geometry, the recovered values of reduced scattering in dermis are overestimated and the recovered values of absorption are underestimated for both analytical models. Presumably, in the presence of a thin mismatched topical layer, only the effective attenuation coefficient of the bottom layer can be accurately recovered using a diffusion theory-based analytical model while separate reconstruction of absorption and reduced scattering fails due to the inapplicability of the method of images. These findings require implementation of more sophisticated models of light transfer in inhomogeneous media in the recovery algorithms.

A sequential method for estimating the optical properties of two-layer agro-products from spatially-resolved diffuse reflectance: Simulation

A sequential method for estimating the optical properties of two-layer biological tissues with spatially-resolved diffuse reflectance was proposed and validated using Monte Carlo simulations. The relationship between the penetration depth of detected photons and source-detector separation was first studied. Photons detected at larger source-detector separations generally penetrated deeper into the medium than those detected at small source-detector separations. The effect of each parameter involved in the two-layer diffusion model (i.e., the absorption and reduced scattering coefficients (μa and μs′) of each layer, and the thickness of top layer) on reflectance was investigated. It was found that the relationship between the optical properties and thickness of top layer was a critical factor in determining whether photons would have sufficient interactions with the top layer and also penetrate into the bottom layer. The constraints for the proposed sequential estimation method were quantitatively determined by the curve fitting procedure coupled with error contour map analyses. Results showed that the optical properties of top layer could be determined within 10% error using the semi-infinite diffusion model for reflectance profiles with properly selected start and end points, when the thickness of top layer was larger than two times its mean free path (mfp′). And the optical properties of the bottom layer could be estimated within 10% error by the two-layer diffusion model, when the thickness of top layer was <16 times its mfp′. The proposed sequential estimation method is promising for improving the estimation of the optical properties of two-layer tissues from the same spatially-resolved reflectance.

Small separation diffuse correlation spectroscopy for measurement of cerebral blood flow in rodents.

Diffuse correlation spectroscopy (DCS) has shown promise as a means to non-invasively measure cerebral blood flow in small animal models. Here, we characterize the validity of DCS at small source-detector reflectance separations needed for small animal measurements. Through Monte Carlo simulations and liquid phantom experiments, we show that DCS error increases as separation decreases, although error remains below 12% for separations > 0.2 cm. In mice, DCS measures of cerebral blood flow have excellent intra-user repeatability and strongly correlate with MRI measures of blood flow (R = 0.74, p<0.01). These results are generalizable to other DCS applications wherein short-separation reflectance geometries are desired.

Performance comparison of the MOXY and PortaMon near-infrared spectroscopy muscle oximeters at rest and during exercise.

The purpose of the study was to compare muscle oxygenation as measured by two portable, wireless near-infrared spectroscopy (NIRS) devices under resting and dynamic conditions. A recently developed low-cost NIRS device (MOXY) was compared against an established PortaMon system that makes use of the spatially resolved spectroscopy algorithm. The influence of increasing external pressure on tissue oxygen saturation index (TSI) indicated that both devices are stable between 2 and 20mmHg. However, above this pressure, MOXY reports declining TSI values. Analysis of adipose tissue thickness (ATT) and TSI shows a significant, nonlinear difference between devices at rest. The devices report similar TSI (%) values at a low ATT (<7 mm) (PortaMon minus MOXY difference is +1.1±2.8%) with the major subsequent change between the devices occurring between 7 and 10mm; at ATT values >10 mm the difference remains constant (-14.7±2.8%). The most likely explanation for this difference is the small source-detector separation (2.5cm) in the MOXY resulting in lower tissue penetration into muscle in subjects with higher ATT. Interday test-retest reliability of resting TSI was evaluated on five separate occasions, with the PortaMon reporting a lower coefficient of variation (1.8% to 2.5% versus 5.7% to 6.2%). In studies on male subjects with low ATT, decreases in the TSI were strongly correlated during isometric exercise, arterial occlusion, and incremental arm crank exercise. However, the MOXY reports a greater dynamic range, particularly during ischemia induced by isometric contraction or occlusion (Δ74.3% versus Δ43.7%; hyperemia MAX-occlusion MIN). This study shows that in this subject group both MOXY and PortaMon produce physiologically credible TSI measures during rest and exercise. However, the absolute values obtained during exercise are generally not comparable between devices unless corrected by physiological calibration following an arterial occlusion.

Multidistance diffuse correlation spectroscopy for simultaneous estimation of blood flow index and optical properties.

Traditionally, diffuse correlation spectroscopy (DCS) measures microvascular blood flow by fitting a physical model to the measurement of the intensity autocorrelation function from a single source-detector pair. This analysis relies on the accurate knowledge of the optical properties, absorption, and reduced scattering coefficients of the medium. Therefore, DCS is often deployed together with diffuse optical spectroscopy. We present an algorithm that employs multidistance DCS (MD-DCS) for simultaneous measurement of bloodflow index, as well as an estimate of the optical properties of the tissue. The algorithm has been validated through noise-free and noise-added simulated data and phantom measurements. A longitudinal in vivo measurement ofa mouse tumor is also shown. MD-DCS is introduced as a stand-alone system for small source-detector separations (<2 cm) for noninvasive measurement of microvascular blood flow.

Analytical Green's function for the fluorescence simplified spherical harmonics equations in turbid medium.

It is more complicated to write the analytical expression for the fluorescence simplified spherical harmonics ( SPN) equations in a turbid medium, since both the processes of the excitation and emission light and the composite moments of the fluence rate are described by coupled equations. Based on an eigen-decomposition strategy and the well-developed analytical methods of diffusion approximation (DA), we derive the analytical solutions to the fluorescence SPN equations for regular geometries using the Green's function approach. By means of comparisons with the results of fluorescence DA and Monte Carlo simulations, we have shown the effectiveness of our proposed method and the expected advantages of the SPN equations in the case of small source-detector separation and high absorption.

Convolution model of the diffuse reflectance for layered tissues

We present an explicit model for the diffuse reflectance due to a collimated beam of light incident normally on layered tissues. This model is derived using the corrected diffusion approximation applied to a layered medium, and it takes the form of a convolution with an explicit kernel and the incident beam profile. This model corrects the standard diffusion approximation over all source-detector separation distances provided the beam is sufficiently wide compared to the scattering mean free path. We validate this model through comparison with Monte Carlo simulations. Then we use this model to estimate the optical properties of an epithelial layer from Monte Carlo simulation data. Using measurements at small source-detector separations and this model, we are able to estimate the absorption coefficient, scattering coefficient, and anisotropy factor of epithelial tissues efficiently with reasonable accuracy.

Rigorous Asymptotic and Moment-Preserving Diffusion Approximations for Generalized Linear Boltzmann Transport in Arbitrary Dimension

We derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation for the stationary collision density and scalar flux about an isotropic point source in an infinite d-dimensional absorbing medium with isotropic scattering. We consider both classical transport theory with exponentially distributed free paths in arbitrary dimensions as well as a number of nonclassical transport theories (nonexponential random flights) that describe a broader class of transport processes within partially correlated random media. New rigorous asymptotic diffusion approximations are derived where possible. We also generalize Grosjean’s moment-preserving approach of separating the first (or uncollided) distribution from the collided portion and approximating only the latter using diffusion. We find that for any spatial dimension and for many free-path distributions Grosjean’s approach produces compact, analytic approximations that are, overall, more accurate for high absorption and for small source-detector separations than either P1 diffusion or rigorous asymptotic diffusion. These diffusion-based approximations are exact in the first two even spatial moments, which we derive explicitly for various nonclassical transport types. We also discuss connections between the random-flight-theory derivation of the Green’s function and the discrete spectrum of the transport operator and report some new observations regarding the discrete eigenvalues of the transport operator for general dimensions and free-path distributions.

Joint derivation method for determining optical properties based on steady-state spatially resolved diffuse reflectance measurement at small source-detector separations and large reduced albedo range: theory and simulation

Accurate determination of the optical properties (the absorption coefficient μ(a) and the reduced scattering coefficient μ(s) (')) of tissues is very important in a variety of diagnostic and therapeutic procedures. Optical diffusion theory is frequently used as the forward model for describing the photon transfer in media with large reduced albedos (a(')) and in large source-detector separations (SDS). Several other methods (PN approximation, hybrid diffusion-P3 approximation) have also been published that describe photon transfer in media with low a(') or small SDSs. We studied the theoretical models for the steady-state spatially resolved diffuse reflectance measurement to accurately determine μ(a) and μ(s) (') at large a(') range but small SDSs. Instead of using a single model, a joint derivation method is proposed. The developed method uses one of the best aforementioned theoretical methods separately in five ranges of a(') determined from several forward models. In the region of small SDSs (the range between 0.4 and 8 mm) and large a(') range (between 0.5 and 0.99), the best theoretical derivation model was determined. The results indicate that the joint derivation method can improve the derivation accuracy and that a(') range can be determined by the steady-state spatially resolved diffuse reflectance measurement.

Sensitivity of Delta-P1 approximation model to second-order parameter

The sensitivity of the second-order parameter of the Henyey-Greenstein phase function based on the spatially-resolved diffuse reflectance within the two-point-source approximation to the Delta-P1 approximation model is studied, and the analytical expression of the sensitivity is derived. The results show that the analytic solution of the Delta-P1 approximation model for reflectance contains the second-order parameter of the scattering phase function compared with the diffusion approximation model, and the second-order parameter has significant influence (the extreme value greater than 30%) on the analytic solution for spatially-resolved diffuse reflectance with small source-detector separations ( 1.5 mm) no matter whether the absorption is weak or strong. The research has theoretical significance for obtaining the optical information abour tissue with the second-order parameter .

Improved recovery of the hemodynamic response in diffuse optical imaging using short optode separations and state-space modeling

Diffuse optical imaging (DOI) allows the recovery of the hemodynamic response associated with evoked brain activity. The signal is contaminated with systemic physiological interference which occurs in the superficial layers of the head as well as in the brain tissue. The back-reflection geometry of the measurement makes the DOI signal strongly contaminated by systemic interference occurring in the superficial layers. A recent development has been the use of signals from small source-detector separation (1cm) optodes as regressors. Since those additional measurements are mainly sensitive to superficial layers in adult humans, they help in removing the systemic interference present in longer separation measurements (3cm). Encouraged by those findings, we developed a dynamic estimation procedure to remove global interference using small optode separations and to estimate simultaneously the hemodynamic response. The algorithm was tested by recovering a simulated synthetic hemodynamic response added over baseline DOI data acquired from 6 human subjects at rest. The performance of the algorithm was quantified by the Pearson R2 coefficient and the mean square error (MSE) between the recovered and the simulated hemodynamic responses. Our dynamic estimator was also compared with a static estimator and the traditional adaptive filtering method. We observed a significant improvement (two-tailed paired t-test, p<0.05) in both HbO and HbR recovery using our Kalman filter dynamic estimator compared to the traditional adaptive filter, the static estimator and the standard GLM technique.

Spectral determination of a two-parametric phase function for polydispersive scattering liquids

A method for determining a two-parametric Gegenbauer-kernel phase function that accurately describes the diffuse reflectance from a polydispersive scattering media at small source-detector separations (0.23 to 1.2 mm), is presented. The method involves spectral collimated transmission measurements, spatially resolved spectral diffuse reflectance (SRDR) measurements, and inverse Monte Carlo technique. Both absolute calibration (using a monodispersive polystyrene microsphere suspension) and relative calibration (eliminating differences between fibers) of SRDR spectra yielded comparable results. When applied to water dilutions of milk, simulated and measured spectra deviated less than 6.5% and 2.5% for the absolute and relative calibration case, respectively, even for the closest fiber separation. Corresponding values for milk including ink as an absorber, were 13.4% and 7.3%.

Modeling of nonphase mechanisms in ultrasonic modulation of light propagation

While phase variation due to ultrasonic modulation of coherent light has been extensively studied in acousto-optical imaging, fewer groups have studied nonphase mechanisms of ultrasonic modulation, which may be important in exploring ultrasonic modulation of incoherent light for imaging. We have developed a versatile Monte Carlo based method that can model not only phase variation due to refractive index changes and scatterer displacement in tissue or tissue-like phantoms, but also amplitude and exit location variations due to the changes in optical properties and refractive index under ultrasonic modulation, in which the exit location variation has not, to the best of our knowledge, been modeled previously. Our results show that the modulation depth due to the exit location variation is three orders of magnitude higher than that due to amplitude variation, but two to three orders of magnitude lower than that due to phase variation for monochromatic light. Furthermore it is found that the modulation depth in reflectance due to the exit location variation is larger than that in transmittance for small source-detector separations.

Time-Resolved Diffuse Reflectance Using Small Source-Detector Separation and Fast Single-Photon Gating

We demonstrate the feasibility of time-resolved diffuse reflectance measurements at small source-detector separations using a single-photon avalanche diode operated in time-gated mode. Photon time distributions at an interfiber distance of 2 mm were obtained on a homogeneous tissue phantom with a dynamic range of 10(6) and collecting photons at arrival times up to 4 ns. Moreover, we were able to detect a local inhomogeneity deeply buried within a diffusive medium with better spatial resolution, higher signal intensity, and same contrast of a larger (20 mm) interfiber distance. Finally, the proposed approach proved valuable to detect in vivo a task-related brain activation.

Analytical model of light reflectance for extraction of the optical properties in small volumes of turbid media

Monte Carlo simulations and experiments in tissue phantoms were used to empirically develop an analytical model that characterizes the reflectance spectrum in a turbid medium. The model extracts the optical properties (scattering and absorption coefficients) of the medium at small source-detector separations, for which the diffusion approximation is not valid. The accuracy of the model and the inversion algorithm were investigated and validated. Four fiber probe configurations were tested for which both the source and the detector fibers were tilted at a predetermined angle, with the fibers parallel to each other. This parallel-fiber geometry facilitates clinical endoscopic applications and ease of fabrication. Accurate extraction of tissue optical properties from in vivo spectral measurements could have potential applications in detecting, noninvasively and in real time, epithelial (pre)cancers.

Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation

We characterize the capability of time-resolved reflectance measurements at small source-detector separation (less than 5 mm) to localize small inhomogeneities embedded in an otherwise homogeneous or layered diffusive medium. By considering both absorption and scattering inhomogeneities, we demonstrate the improvement of this approach in terms of contrast and spatial resolution, as compared to more typical set-ups involving larger source-detection separations (few centimeters). Simulations are performed exploiting an analytical perturbation approach to diffusion theory and a four-layer heterogeneous time-resolved Monte Carlo code, considering realistic tissue geometries. Exhaustive investigation in the parameters space is reported.

Determination of reduced scattering coefficient of biological tissue from a needle-like probe

Detection of interactions between light and tissue can be used to characterize the optical properties of the tissue. The purpose of this paper is to develop an algorithm that determines the reduced scattering coefficient (mus') of tissues from a single optical reflectance spectrum measured with a small source-detector separation. A qualitative relationship between mus' and optical reflectance was developed using both Monte Carlo simulations and empirical tissue calibrations for each of two fiber optic probes with 400-mum and 100-mum fibers. Optical reflectance measurements, using a standard frequency-domain oximeter, were performed to validate the calculated mus' values. The algorithm was useful for determining mus' values of in vivo human fingers and rat brain tissues.

Sampling tissue volumes using frequency-domain photon migration.

We demonstrate the use of Monte Carlo simulations to generate photon scattering density functions (PSDFs) that represent the tissue volume sampled by steady-state and frequency-domain photon migration. We use these results to illustrate how scaling laws can be developed to determine the mean sampling depth of the multiply scattered photons detected by photon migration methods that remain valid outside the bounds of the standard diffusion approximation, i.e., at small source-detector separations and in media where the optical absorption is significant relative to scattering. Using both the PSDF computation and the newly formulated scaling laws, we focus on a comprehensive description of the effects of source modulation frequency, optical absorption, and source-detector separation on the depth of the sampled tissue volume as well as the sensitivity of frequency-domain photon migration measurements to the presence of a localized absorption heterogeneity.

Small-volume frequency-domain oximetry: phantom experiments and first in vivo results

We describe a new method to determine the oxygen saturation and the total hemoglobin content of tissue in vivo absolutely at small source-detector separations (<10 mm). Phase and mean intensity of modulated laser light of various wavelengths was measured at several predetermined source-detector separations in the frequency domain. From these measured quantities, the absorption coefficient was derived using the modified time-integrated microscopic Beer-Lambert law (MBL). In addition, the interaction volume of the photons was determined using a multi-layer Monte-Carlo model of human skin. To evaluate the method, we employed homogenous solid phantoms (consisting of TiO2 particles embedded in resin) with mean scattering and absorbing properties comparable to those of human skin. Furthermore, in vivo measurements were performed in a healthy volunteer to demonstrate that the technique is applicable for the determination of the oxygen saturation and the total hemoglobin content in the skin in vivo. The proposed technique is especially suited for the on-line determination of the oxygen saturation and total hemoglobin content in applications where small applicators are required (e.g., fetal oxygen monitoring sub partu).

Absolute absorber quantification in turbid media at small source–detector separations

The aim of our studies was to develop a method to determine the absorption coefficient of a turbid medium in a reflection geometry with small source–detector separations. Therefore, the time-integrated microscopic Beer–Lambert law (MBL) was modified in order to obtain the absolute absorption coefficient from mean time of flight and dc-intensity measurements. The new technique was evaluated using turbid phantoms having varying scattering (μs′ between 0.2 and 2.2 mm-1) and absorbing properties (μa between 0.04 and 0.14 mm-1), comparable to many biological tissues at various source–detector separations between 3 and 11 mm. The measurements were performed at a wavelength of 1064 nm. We found that this new method was able to determine the absolute absorption coefficient of the selected phantoms with a standard error of less than 0.005 mm-1 over the range of optical properties investigated.