Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation, and sensitivity analysis

Retrieval of chromophore concentration changes in a digital human head model using analytical mean partial pathlengths of photons

Significance

Continuous-wave functional near-infrared spectroscopy has proved to be a valuable tool for assessing hemodynamic activity in the human brain in a non-invasively and inexpensive way. However, most of the current processing/analysis methods assume the head is a homogeneous medium, and hence do not appropriately correct for the signal coming from the scalp. This effect can be reduced by considering light propagation in a layered model of the human head, being the Monte Carlo (MC) simulations the gold standard to this end. However, this implies large computation times and demanding hardware capabilities.

Aim

In this work, we study the feasibility of replacing the homogeneous model and the MC simulations by means of analytical multilayered models, combining in this way, the speed and simplicity of implementation of the former with the robustness and accuracy of the latter.

Approach

Oxy- and deoxyhemoglobin (HbO and HbR, respectively) concentration changes were proposed in two different layers of a magnetic resonance imaging (MRI)-based meshed model of the human head, and then these changes were retrieved by means of (i) a typical homogeneous reconstruction and (ii) a theoretical layered reconstruction.

Results

Results suggest that the use of analytical models of light propagation in layered models outperforms the results obtained using traditional homogeneous reconstruction algorithms, providing much more accurate results for both, the extra- and the cerebral tissues. We also compare the analytical layered reconstruction with MC-based reconstructions, achieving similar degrees of accuracy, especially in the gray matter layer, but much faster (between 4 and 5 orders of magnitude).

Conclusions

We have successfully developed, implemented, and validated a method for retrieving chromophore concentration changes in the human brain, combining the simplicity and speed of the traditional homogeneous reconstruction algorithms with robustness and accuracy much more similar to those provided by MC simulations.

Reconstruction of light absorption changes in the human head using analytically computed photon partial pathlengths in layered media

Functional near infrared spectroscopy has been used in recent decades to sense and quantify changes in hemoglobin concentrations in the human brain. This noninvasive technique can deliver useful information concerning brain cortex activation associated with different motor/cognitive tasks or external stimuli. This is usually accomplished by considering the human head as a homogeneous medium; however, this approach does not explicitly take into account the detailed layered structure of the head, and thus, extracerebral signals can mask those arising at the cortex level. This work improves this situation by considering layered models of the human head during reconstruction of the absorption changes in layered media. To this end, analytically calculated mean partial pathlengths of photons are used, which guarantees fast and simple implementation in real-time applications. Results obtained from synthetic data generated by Monte Carlo simulations in two- and four-layered turbid media suggest that a layered description of the human head greatly outperforms typical homogeneous reconstructions, with errors, in the first case, bounded up to ∼20% maximum, while in the second case, the error is usually larger than 75%. Experimental measurements on dynamic phantoms support this conclusion.

Optical property recovery with spatially-resolved diffuse reflectance at short source-detector separations using a compact fiber-optic probe

We describe a compact fiber-optic probe (2 mm outside diameter) that utilizes spatially-resolved diffuse reflectance for tissue optical property recovery. Validation was performed in phantoms containing Intralipid 20% as scatterer, and methylene blue (MB), MnTPPS, and/or India ink as absorbers. Over a range of conditions, the reduced scattering coefficient was recovered with a root mean square error (RMSE) of 0.86-2.7 cm^-1 (average error = 3.8%). MB concentration was recovered with RMSE = 0.26-0.52 µM (average error = 15.0%), which did not vary with inclusion of MnTPPS (p=0.65). This system will be utilized to determine optical properties in human abscesses, in order to generate treatment plans for photodynamic therapy.

Simultaneous recovery of a full set of optical properties in turbid media using incomplete P5 approximation to CW radiance

Traditional methods for determining the optical properties of turbid media are implemented in diffusive regimes, where the scattering coefficient μ_s and anisotropy factor g are inseparable due to the similarity relation for anisotropy scattering. Determining μ_s and g in addition to the absorption coefficient μ_a normally requires measurements using thin samples or media of low scattering, which are inappropriate for in vivo applications. In this work, we propose an analytical method to simultaneously recover μ_a, μ_s, and g based on the incomplete P5 approximation (P5_in) to the continuous wave (CW) radiative transport equation. The proposed method was verified using both simulated and experimental data with the relative errors less than 6.6%, 11.6%, and 8.2% for μ_a ranging 0.0071-0.0168  mm^-1, μ_s ranging 2.35-8.47  mm^-1, and g ranging 0.61-0.81, respectively. Since the P5_in-based radiance method can be easily implemented with several measurements, it is expected to be used for recovering a full set of the optical properties in vivo.

Determination of optical properties of turbid medium from relative interstitial CW radiance measurements using the incomplete P3 approximation

Interstitial determination of the tissue optical properties is important in biomedicine, especially for interstitial laser therapies. Continuous wave (CW) radiance techniques which examine light from multiple directions have been proposed as minimally invasive methods for determining the optical properties under an interstitial probe arrangement. However, both the fitting algorithm based on the P3 approximation and the analytical method based on the diffusion approximation (DA), which are currently used recovery algorithms, cannot extract the optical properties of tissue with low transport albedos accurately from radiance measurements. In this paper, we proposed an incomplete P3 approximation for the radiance, the P3_in for short, which is the asymptotic part of the solution for the P3 approximation. The relative differences between the P3_in and the P3 were within 0.48% over a wide range of clinically relevant optical properties for measurements at source detector separations (SDS) from 5 mm to 10 mm and angles from 0° to 160°. Based on the P3_in, we developed an analytical method for extracting the optical properties directly using simple expressions constructed from the radiance measurements at only two SDSs and four angles. The developed recovery algorithm was verified by simulated and experimental radiance data. The results show that both the absorption and reduced scattering coefficients were recovered accurately with relative errors within 5.28% and 3.86%, respectively, from the simulated data and with relative errors within 10.82% and 10.67%, respectively, from the experimental data over a wide range of albedos from 0.5 to 0.99. Since the developed P3_in-based radiance technique can obtain the optical properties rapidly from the measurements at only two SDSs and four angles, it is expected to be used for in vivo and in situ determination of the optical properties in online treatment planning during laser therapies.

Recovery of optical properties using interstitial cylindrical diffusers as source and detector fibers

We demonstrate recovery of optical properties using arrays of interstitial cylindrical diffusing fibers as sources and detectors. A single 1-cm diffuser delivered laser illumination at 665 nm, while seven 1- and 2-cm diffusers at 1-cm grid spacing acted as detectors. Extraction of optical properties from these measurements was based upon a diffusion model of emission and detection distributions for these diffuser fibers, informed by previous measurements of heterogeneous axial detection. Verification of the technique was performed in 15 liquid tissue-simulating phantoms consisting of deionized water, India ink as absorber, and Intralipid 20% as scatterer. For the range of optical properties tested, mean errors were 4.4% for effective attenuation coefficient, 12.6% for absorption coefficient, and 7.6% for reduced scattering coefficient. Error in recovery tended to increase with decreasing transport albedo. For therapeutic techniques involving the delivery of light to locations deep within the body, such as interstitial photodynamic and photothermal therapies, the methods described here would allow the treatment diffuser fibers also to be used as sources and detectors for recovery of optical properties. This would eliminate the need for separately inserted fibers for spectroscopy, reducing clinical complexity and improving the accuracy of treatment planning.

Implementation of laser acupuncture with lifting-thrusting through the use of mechanically immobile components

Lifting and thrusting constitute an important manipulation method in traditional Chinese acupuncture. Lifting and thrusting enables the implementation of various features, such as reinforcement and reduction, which enhance acupuncture effectiveness. Laser acupuncture stimulates acupoints through laser light, which is a noninvasive treatment, but can still achieve effects similar to those obtained from traditional acupuncture. Lifting and thrusting can be achieved by moving the laser focal point back and forth, thus concentrating the energy, as does the tip of the acupuncture needle when it is moved upward and downward in the acupoint. This article presents a novel optical design of a laser acupuncture device, in which a focus-tunable lens is used to move the position of the focused light in order to achieve the lifting and thrusting mechanism through programmable changes to the control current of the focus-tunable lens. The device employs an infrared laser with a wavelength of 808 nm and a maximum power of 150 mW. The focus-tunable lens used in this study had a diopter of -10 to +5. The results revealed that by controlling the lens diopter, the focused light can be moved from 4.5 to 9.5 cm. Therefore, the range of the lift and thrust for the laser acupuncture device was 5 cm. The area of the focal point was approximately 6×10(-3) mm2, which is comparable to that of the commonly used traditional acupuncture needle tip. Because the components are immobile, no additional space is required for the moving lens. Therefore, the size of the laser acupuncture head can be minimized, and the effectiveness of focus tuning can be improved.

Tagging photons with gold nanoparticles as localized absorbers in optical measurements in turbid media

We analyze a role of a localized inclusion as a probe for spatial distributions of migrating photons in turbid media. We present new experimental data and two-dimensional analysis of radiance detection of a localized absorptive inclusion formed by gold nanoparticles in Intralipid-1% when the target is translated along the line connecting the light source and detector. Data are analyzed using the novel analytical expression for the relative angular photon distribution function for radiance developed by extending the perturbation approach for fluence. Obtained photon maps allow predicting conditions for detectability of inclusions for which proximity to the detector is essential.

Determination of optical properties by interstitial white light spectroscopy using a custom fiber optic probe

We demonstrate interstitial recovery of absorption and scattering coefficients using a custom optical probe and a Monte Carlo (MC)-based recovery algorithm. The probe consists of six side-firing spectroscopy fibers contained in a 1.1-mm outer diameter cladding, with each fiber having a different axial and angular position on the probe. Broadband white light is delivered by one of the fibers and is detected steady-state by the remaining fibers. These spatially and spectrally resolved data are analyzed using a MC-based fitting algorithm in order to extract the local optical properties. The technique was verified in tissue-simulating phantoms consisting of Intralipid-20% as a scatterer and either manganese meso-tetra (4-sulfanatophenyl) porphine or intact human erythrocytes as an absorber. Absorption coefficients were recovered with a mean error of 9% and scattering coefficients were recovered with a mean error of 19%, whereas the hemoglobin oxygen saturation was recovered with a mean error of 12%. These results demonstrate the feasibility of optical property recovery for situations in which surface-contact spectroscopy is not a possibility, and where only a single probe can be inserted into the tissue.

Radiative transport produced by oblique illumination of turbid media with collimated beams

We examine the general problem of light transport initiated by oblique illumination of a turbid medium with a collimated beam. This situation has direct relevance to the analysis of cloudy atmospheres, terrestrial surfaces, soft condensed matter, and biological tissues. We introduce a solution approach to the equation of radiative transfer that governs this problem, and develop a comprehensive spherical harmonics expansion method utilizing Fourier decomposition (SHEF(N)). The SHEF(N) approach enables the solution of problems lacking azimuthal symmetry and provides both the spatial and directional dependence of the radiance. We also introduce the method of sequential-order smoothing that enables the calculation of accurate solutions from the results of two sequential low-order approximations. We apply the SHEF(N) approach to determine the spatial and angular dependence of both internal and boundary radiances from strongly and weakly scattering turbid media. These solutions are validated using more costly Monte Carlo simulations and reveal important insights regarding the evolution of the radiant field generated by oblique collimated beams spanning ballistic and diffusely scattering regimes.

Separation of absorption and scattering properties of turbid media using relative spectrally resolved cw radiance measurements

We present a new method for extracting the effective attenuation coefficient and the diffusion coefficient from relative spectrally resolved cw radiance measurements using the diffusion approximation. The method is validated on both simulated and experimental radiance data sets using Intralipid-1% as a test platform. The effective attenuation coefficient is determined from a simple algebraic expression constructed from a ratio of two radiance measurements at two different source-detector separations and the same 90° angle. The diffusion coefficient is determined from another ratio constructed from two radiance measurements at two angles (0° and 180°) and the same source-detector separation. The conditions of the validity of the method as well as possible practical applications are discussed.

Infinite space Green's function of the time-dependent radiative transfer equation

This study contains the derivation of an infinite space Green's function of the time-dependent radiative transfer equation in an anisotropically scattering medium based on analytical approaches. The final solutions are analytical regarding the time variable and given by a superposition of real and complex exponential functions. The obtained expressions were successfully validated with Monte Carlo simulations.

Analytical Green's function of the radiative transfer radiance for the infinite medium

An analytical solution of the radiative transfer equation for the radiance caused by an isotropic source which is located in an infinitely extended medium was derived using the P{N} method. The results were compared with Monte Carlo simulations and excellent agreement was found. In addition, the radiance of the SP{N} approximation for the same geometry was derived. Comparison with Monte Carlo simulations showed that the SP{N} radiance, although being more exact than the radiance derived from diffusion theory, has relatively large errors in many relevant cases.

A fiberoptic reflectance probe with multiple source-collector separations to increase the dynamic range of derived tissue optical absorption and scattering coefficients

Measurement of tissue optical absorption and (transport) reduced scattering coefficients (mu(a) and mu(s)', respectively) is fundamental to many applications of light in medicine and biology. We report a handheld fiberoptic probe to determine these coefficients by measuring the diffuse reflectance at multiple source-collector distances, which allows for a larger dynamic range than a single source-collector separation. Diffusion theory and a priori knowledge of the spectral shape of mu(a) and mu(s)' are used in a forward model of the diffuse reflectance. The dynamic range and accuracy of this method were evaluated using Monte Carlo simulations, phantom experiments and tissues in vivo.

Frequency domain measurements on turbid media with strong absorption using the PN approximation

We have applied the frequency-domain technique to measurement of the optical properties of turbid media with strong absorption in the infinite medium limit. Absorption coefficients up to 2.3 cm(-1) for a modified scattering coefficient of 4.3 cm(-1) are studied, which corresponds to a reduced scattering albedo of 0.65. Low phase noise and good phase stability are required for these low albedo conditions. As the degree of absorption increases, the phase changes are reduced while amplitude changes increase. For this reason, correction of amplitude-phase cross talk is essential to achieve accurate measurements with strong absorption. Careful control of stray reflections is required to properly measure amplitude-phase cross talk. Because the diffusion approximation becomes less accurate, measurements are compared to calculations performed in the PN approximation, which is essentially an exact solution for the infinite medium limit. Agreement between theory and experiment is only obtained when correction for amplitude-phase cross talk is performed. These measurements can provide a good method for testing amplitude-phase cross talk.

Frequency domain photothermoacoustic signal amplitude dependence on the optical properties of water: turbid polyvinyl chloride-plastisol system

Photoacoustic (more precisely, photothermoacoustic) signals generated by the absorption of photons can be related to the incident laser fluence rate. The dependence of frequency domain photoacoustic (FD-PA) signals on the optical absorption coefficient (micro(a)) and the effective attenuation coefficient (micro(eff)) of a turbid medium [polyvinyl chloride-plastisol (PVCP)] with tissuelike optical properties was measured, and empirical relationships between these optical properties and the photoacoustic (PA) signal amplitude and the laser fluence rate were derived for the water (PVCP system with and without optical scatterers). The measured relationships between these sample optical properties and the PA signal amplitude were found to be linear, consistent with FD-PA theory: micro(a)=a(A/Phi)-b and micro(eff)=c(A/Phi)+d, where Phi is the laser fluence, A is the FD-PA amplitude, and a, ...,d are empirical coefficients determined from the experiment using linear frequency-swept modulation and a lock-in heterodyne detection technique. This quantitative technique can easily be used to measure the optical properties of general turbid media using FD-PAs.