Photon path distribution and optical responses of turbid media: theoretical analysis based on the microscopic Beer-Lambert law

Beer-Lambert law for optical tissue diagnostics: current state of the art and the main limitations

SIGNIFICANCE

Beer-Lambert law (BLL) is a widely used tool for contact and remote determination of absorber concentration in various media, including living tissues. Originally proposed in the 18th century as a simple exponential expression, it has survived numerous modifications and updates. The basic assumptions of this law may not be fulfilled in real measurement conditions. This can lead to mistaken or misinterpreted results. In particular, the effects to be additionally taken into account in the tissue measurements include anisotropy, scattering, fluorescence, chemical equilibria, interference, dichroism, spectral bandwidth disagreements, stray radiation, and instrumental effects.

AIM

We review the current state of the art and the main limitations of remote tissue diagnostics using the BLL. Historical development of updating this law by taking into account specific additional factors such as light scattering and photon pathlengths in diffuse reflectance is described, along with highlighting the main risks to be considered by interpreting the measured data.

APPROACH

Literature data related to extension and modification of the BLL related to tissue assessment and concentration estimation of specific tissue molecules are collected and analyzed. The main emphasis here is put on the optical measurements of living tissue chromophore concentrations and estimation of physiological parameters, e.g., blood oxygen saturation.

RESULTS

Modified expressions of the BLL suitable for several specific cases of living tissue characterization are presented and discussed.

CONCLUSIONS

Applications of updated/modified Beer-Lambert law (MBLL) with respect to particular measurement conditions are helpful for obtaining more reliable data on the target tissue physiological state and biochemical content. MBLL accounting for the role of scattering in several ways appears to be a successful approach. Extended MBLL and BLL in the time domain form could provide more accurate results, but this requires more time resources to be spent.

Time-Gated Single-Photon Detection in Time-Domain Diffuse Optics: A Review

This work reviews physical concepts, technologies and applications of time-domain diffuse optics based on time-gated single-photon detection. This particular photon detection strategy is of the utmost importance in the diffuse optics field as it unleashes the full power of the time-domain approach by maximizing performances in terms of contrast produced by a localized perturbation inside the scattering medium, signal-to-noise ratio, measurement time and dynamic range, penetration depth and spatial resolution. The review covers 15 years of theoretical studies, technological progresses, proof of concepts and design of laboratory systems based on time-gated single-photon detection with also few hints on other fields where the time-gated detection strategy produced and will produce further impact.

Time-Domain Near-Infrared Spectroscopy and Imaging: A Review

This article reviews the past and current statuses of time-domain near-infrared spectroscopy (TD-NIRS) and imaging. Although time-domain technology is not yet widely employed due to its drawbacks of being cumbersome, bulky, and very expensive compared to commercial continuous wave (CW) and frequency-domain (FD) fNIRS systems, TD-NIRS has great advantages over CW and FD systems because time-resolved data measured by TD systems contain the richest information about optical properties inside measured objects. This article focuses on reviewing the theoretical background, advanced theories and methods, instruments, and studies on clinical applications for TD-NIRS including some clinical studies which used TD-NIRS systems. Major events in the development of TD-NIRS and imaging are identified and summarized in chronological tables and figures. Finally, prospects for TD-NIRS in the near future are briefly described.

Effect of probe geometry and optical properties on the sampling depth for diffuse reflectance spectroscopy

The sampling depth of light for diffuse reflectance spectroscopy is analyzed both experimentally and computationally. A Monte Carlo (MC) model was used to investigate the effect of optical properties and probe geometry on sampling depth. MC model estimates of sampling depth show an excellent agreement with experimental measurements over a wide range of optical properties and probe geometries. The MC data are used to define a mathematical expression for sampling depth that is expressed in terms of optical properties and probe geometry parameters.

Algorithm for automated selection of application-specific fiber-optic reflectance probes

Several optical techniques and fiber-optic probe systems have been designed to measure the optical properties of tissue. While a wide range of options is often beneficial, it poses a problem to investigators selecting which method to use for their biomedical application of interest. We present a methodology to optimally select a probe that matches the application requirements. Our method is based both on matching a probe's mean sampling depth with the optimal diagnostic depth of the clinical application and on choosing a probe whose interrogation depth and path length is the least sensitive to alterations in the target medium's optical properties. Satisfying these requirements ensures that the selected probe consistently assesses the relevant tissue volume with minimum variability. To aid in probe selection, we have developed a publicly available graphical user interface that takes the desired sampling depth and optical properties of the medium as its inputs and automatically ranks different techniques in their ability to robustly target the desired depth. Techniques investigated include single fiber spectroscopy, differential path length spectroscopy, polarization-gating, elastic light scattering spectroscopy, and diffuse reflectance. The software has been applied to biological case studies.

Analytical light reflectance models for overlapping illumination and collection area geometries

Several biomedical applications, such as detection of dysplasia, require selective interrogation of superficial tissue structures less than a few hundred micrometers thick. Techniques and methods have been developed to limit the penetration depth of light in tissue, including the design of systems such as fiber-optic probes that have overlapping illumination and collection areas on the tissue surface. For such geometries, the diffusion approximation to the light-transport equation typically does not apply, and as a result there is no general model to extract tissue optical properties from reflectance measurements. In the current study, we employ Monte Carlo (MC) simulations to develop simple and compact analytical models for the light reflectance from these overlapping geometries. These models incorporate the size of the illumination and collection areas, the collection angle, the polarization of the incident light, and the optical properties of the sample. Moreover, these MC simulations use the Whittle-Matérn model to describe scattering from spatially continuous refractive index media such as tissue, which is more general than models based on the conventionally used Henyey-Greenstein model. We validated these models on tissue-simulating phantoms. The models developed herein will facilitate the extraction of optical properties and aid in the design of optical systems employing overlapping illumination and collection areas, including fiber-optic probes for in vivo tissue diagnosis.

Quantitation of the hydroxycinnamic acid derivatives and the glycosides of flavonols and flavones by UV absorbance after identification by LC-MS

A general approach was developed to quantify hydroxycinnamic acid derivatives and the glycosides of flavonols and flavones using UV molar relative response factors (MRRFs). More than 90 standards were analyzed by LC-MS and divided into five groups based on the λ(max) of their band I absorbance profiles. For each group, a commercially available standard was chosen as the group reference standard. Response factors were determined for each standard in each group as purchased (MRRF) and, when possible, after vacuum drying (MRRF(D)). The MRRF(D) values for 17 compounds whose λ(max) values fell within ±2 nm of the group reference standard were 1.01 ± 0.03. MRRF values for compounds whose λ(max) values fell within ±10 nm of the group reference standard were 0.96 ± 0.13. Group reference standards were used to quantify 44 compounds in Chinese lettuce, red onion, and white tea. This approach allows quantitation of numerous compounds for which there are no standards.

Time-resolved optical mammography and its preliminary clinical results

We have been developing an optical mammography prototype consisting of a multi-channel time-resolved spectroscopy system for breast cancer screening. The system utilizes the time-correlated single photon counting method, and the detector modules and the signal processing circuits were custom-made to obtain a high signal to noise ratio and high temperature stability with a high temporal resolution. Pulsed light generated by a Ti: Sapphire laser was irradiated to the breast, and the transmitted light was collected by optical fibers placed on the surface of a hemispherical gantry filled with an optical matching fluid. To reconstruct a 3D image of the breast, we employed a method using a time-resolved photon path distribution based on the assumption that scattering and absorption are independent of each other. We verified the possibility of human breast imaging by using a three-dimensional phantom model, which provides a simulation of human breast cancer, in the gantry. The clinical study was also started in January 2007. In a comparative study with conventional modalities, the breast cancers were detected as regions of optically higher absorption. Moreover, the results suggest that optical mammography is useful in monitoring the effects of chemotherapy.

Simulation of near-infrared light absorption considering individual head and prefrontal cortex anatomy: implications for optical neuroimaging

Functional near-infrared spectroscopy (fNIRS) is an established optical neuroimaging method for measuring functional hemodynamic responses to infer neural activation. However, the impact of individual anatomy on the sensitivity of fNIRS measuring hemodynamics within cortical gray matter is still unknown. By means of Monte Carlo simulations and structural MRI of 23 healthy subjects (mean age: 25.0±2.8 years), we characterized the individual distribution of tissue-specific NIR-light absorption underneath 24 prefrontal fNIRS channels. We, thereby, investigated the impact of scalp-cortex distance (SCD), frontal sinus volume as well as sulcal morphology on gray matter volumes (V(gray)) traversed by NIR-light, i.e. anatomy-dependent fNIRS sensitivity. The NIR-light absorption between optodes was distributed describing a rotational ellipsoid with a mean penetration depth of (23.6±0.7) mm considering the deepest 5% of light. Of the detected photon packages scalp and bone absorbed (96.4±9.7)% and V(gray) absorbed (3.1±1.8)% of the energy. The mean V(gray) volume (1.1±0.4) cm3 was negatively correlated (r=-.76) with the SCD and frontal sinus volume (r=-.57) and was reduced by 41.5% in subjects with relatively large compared to small frontal sinus. Head circumference was significantly positively correlated with the mean SCD (r=.46) and the traversed frontal sinus volume (r=.43). Sulcal morphology had no significant impact on V(gray). Our findings suggest to consider individual SCD and frontal sinus volume as anatomical factors impacting fNIRS sensitivity. Head circumference may represent a practical measure to partly control for these sources of error variance.

CT imaging of diffuse medium by time-resolved measurement of backscattered light

Backscattered light was used to reconstruct cross-sectional images of absorption distributions in diffuse media. For efficient and accurate reconstruction, the inverse problem was solved for one dimension, thereby yielding the absorption distribution in a depth direction. A cross-sectional image or three-dimensional structure is reconstructed by shifting a source-detector pair along the object surface. The object is divided into imaginary layers to solve the inverse problem. This solution's accuracy is further improved by solving the problem for two groups of layers successively instead of solving for all layers simultaneously. The technique's effectiveness was verified using solid phantoms and biological tissues.

Significance of background optical properties, time-resolved information and optode arrangement in diffuse optical imaging of term neonates

The significance of accurate knowledge of background optical properties and time-resolved information in reconstructing images of hemodynamic changes in the neonatal brain from diffuse optical imaging data was studied using Monte Carlo (MC) simulation. A segmented anatomical magnetic resonance (MR) image and literature-derived optical properties for each tissue type were used to create a voxel-based anatomical model. Small absorbing perturbations were introduced into the anatomical model to simulate localized hemodynamic responses related to brain activation. Perturbation MC (pMC) was used as the primary method of image reconstruction. For comparison, reconstructions were also performed using the finite element method (FEM) to solve the diffusion approximation (DA) to the radiative transfer equation (RTE). The effect of optode layout was investigated using three different grids. Of the factors studied, the density of the optode grid was found to have the greatest effect on image quality. The use of time-resolved information significantly improved the spatial accuracy with all optode grids. Adequate knowledge and modeling of the optical properties of the background was found to significantly improve the spatial accuracy of the reconstructed images and make the recovery of contrast of absorption changes more consistent over simplified modeling. Localization accuracy of small perturbations was found to be 2-3 mm with accurate a priori knowledge of the background optical properties, when a grid with high optode density (>1 optode cm(-2)) was used.

Reduction of error in spectrophotometry of scattering media using polarization techniques

Scattering can result in erroneous determination of the concentrations of constituent absorbers in spectrophotometry. This is due to the relationship between attenuation and absorption coefficient becoming nonlinear; hence, the use of the Lambert-Beer law becomes invalid. It has previously been shown that application of polarization techniques can reduce these effects, resulting in a more linear relationship. Here we quantify the impact of this improvement on measurement of the ratio of concentrations for two general absorbing species and show that measurement using polarization-maintaining light is more accurate. This is performed using a generalized version of theory previously dependent on selection of isosbestic wavelengths. For the absorbing species and geometries considered here, the mean error on the estimation of absorber concentration ratio is 18.2% for the case of detection without polarization discrimination. When polarization-maintaining light is extracted, mean errors of 1.2% and 5.1% are achieved for linear and circular polarizations, respectively. The improvement provided by the polarization techniques is observed regardless of the illuminating wavelengths but is achieved at the expense of a reduced signal-to-noise ratio. Taking this into account, for the detection scheme considered with a detector well capacity of 4 x 10(5) electrons the improvement provided by linear polarization-maintaining light is reduced to a factor of 3.6 and for circular polarizations a factor of 2.2.

Quantitative spectrophotometry of scattering media via frequency-domain and constant-intensity measurements

The attenuation of light by scattering and absorbing media is nonlinearly dependent upon the absorption coefficient, since detected light has experienced many different flight times. The frequency response of such a sample to modulated light is also nonlinear. We derive an expression for the attenuation that includes both the absorption coefficient and the modulation frequency. Its form is a power series whose coefficients are the cumulants of the temporal point-spread function (TPSF). Recasting this expression in terms of intensity leads to a similar expression with the cumulants replaced by the moments of the TPSF. A means of exploiting this relationship to produce estimates of the absolute concentration of the absorbing species is suggested.

Individual value of brain tissue oxygen pressure, microvascular oxygen saturation, cytochrome redox level, and energy metabolites in detecting critically reduced cerebral energy state during acute changes in global cerebral perfusion

The authors assessed the diagnostic value of brain tissue oxygen tension (PbrO2), microvascular oxygen saturation (SmvO2), cytochrome oxidase redox level (Cyt a+a3 oxidation), and cerebral energy metabolite concentrations in detecting acute critical impairment of cerebral energy homeostasis. Each single parameter as well as derived multimodal indices (arteriovenous difference in oxygen content [AVDO2], cerebral metabolic rate for oxygen [CMRO2], fractional microvascular oxygen extraction [OEF]) were investigated during controlled variation of global cerebral perfusion using a cisternal infusion technique in 16 rabbits. The objective of this study was to determine whether acute changes between normal, moderately, and critically reduced cerebral perfusion as well as frank ischemia defined by local cortical blood flow (lcoBF), brain electrical activity (BEA), and brain stem vasomotor control can be reliably identified by SmvO2, PbrO2, Cyt a+a3 oxidation, or energy metabolites (glutamate, lactate/pyruvate ratio). PbrO2, SmvO2, and Cyt a+a3 oxidation, but not cerebral perfusion pressure, were closely linked to lcoBF and BEA and allowed discrimination between normal, moderately reduced, and critically reduced cerebral perfusion (P < 0.01). Glutamate concentrations and the lactate/pyruvate ratio varied significantly only between moderately reduced cerebral perfusion and frank ischemia (complete loss of BEA and brain stem vasomotor control). Therefore, PbrO2, SmvO2, and Cyt a+a3 oxidation, but not glutamate and the lactate/pyruvate ratio, reliably predict the transition from moderately to critically reduced cerebral perfusion with impending energy failure.

Penetration depth of light re-emitted by a diffusive medium: theoretical and experimental investigation

The depth at which photons penetrate into a diffusive medium before being re-emitted has been investigated with reference to a semi-infinite homogeneous medium illuminated by a pencil beam. By using the diffusion equation analytical expressions have been obtained for the probability that photons penetrate at a certain depth before being detected, and for the mean path length they travel inside each layer of the medium. Expressions have been obtained both for the cw and the time domain, and simple approximate scaling relationships describing the dependence on the scattering properties of the medium have been found. For time-resolved measurements both the probability and the mean path length are expected to be independent of the distance from the light beam at which the detector is placed and of the absorption coefficient of the medium. The penetration depth increases as the time of flight increases. In contrast, for cw measurements both the probability and the mean path length strongly depend on the distance and absorption. The penetration depth increases as the distance increases or absorption decreases. The accuracy of the analytical expressions has been demonstrated by comparisons with cw experimental results. The penetration depth and the mean path length provide useful information, for instance, for measurements of tissue oxygenation and for functional imaging of muscle and brain. In particular, the depth reached by received photons provides overall information on the volume of the tissue actually investigated, while the mean path is strictly related to the sensitivity to local variations of absorption.

Photon path distribution in inhomogeneous turbid media: theoretical analysis and a method of calculation

The photon path distribution (PPD) is a measure that I have developed to express optical responses in inhomogeneous turbid media in the time and frequency domains. The PPD is defined by local photon pathlengths of possible photons having total zigzag pathlengths I between the points of light input and detection. Such a distribution is independent of absorption and is uniquely determined for the medium under quantification. I show that the PPD is derived through the local photon count of the possible photons arising from an optical impulse incident on an imaginary medium having the same optical properties as the medium under quantification, except for the absence of absorption. The formulas derived can be used to calculate the PPD simultaneously with, for example, the numerical calculation of a diffusion equation.