Application of the Laplace transform in time-domain optical spectroscopy and imaging

Two-layer reconstruction of Raman spectra in diffusive media based on an analytical model in the time domain

We derive and validate an analytical model that describes the migration of Raman scattered photons in two-layer diffusive media, based on the diffusion equation in the time domain. The model is derived under a heuristic approximation that background optical properties are identical on the excitation and Raman emission wavelengths. Methods for the reconstruction of two-layer Raman spectra have been developed, tested in computer simulations and validated on tissue-mimicking phantom measurements data. Effects of different parameters were studied in simulations, showing that the thickness of the top layer and number of detected photon counts have the most significant impact on the reconstruction. The concept of quantitative, mathematically rigorous reconstruction using the proposed model was finally proven on experimental measurements, by successfully separating the spectra of silicone and calcium carbonate (calcite) layers, showing the potential for further development and eventual application in clinical diagnostics.

Two-layered blood-lipid phantom and method to determine absorption and oxygenation employing changes in moments of DTOFs

Near-infrared spectroscopy (NIRS) is an established technique for measuring tissue oxygen saturation (StO₂), which is of high clinical value. For tissues that have layered structures, it is challenging but clinically relevant to obtain StO₂ of the different layers, e.g. brain and scalp. For this aim, we present a new method of data analysis for time-domain NIRS (TD-NIRS) and a new two-layered blood-lipid phantom. The new analysis method enables accurate determination of even large changes of the absorption coefficient (Δµ_a) in multiple layers. By adding Δµ_a to the baseline µ_a, this method provides absolute µ_a and hence StO₂ in multiple layers. The method utilizes (i) changes in statistical moments of the distributions of times of flight of photons (DTOFs), (ii) an analytical solution of the diffusion equation for an N-layered medium, (iii) and the Levenberg-Marquardt algorithm (LMA) to determine Δµ_a in multiple layers from the changes in moments. The method is suitable for NIRS tissue oximetry (relying on µ_a) as well as functional NIRS (fNIRS) applications (relying on Δµ_a). Experiments were conducted on a new phantom, which enabled us to simulate dynamic StO₂ changes in two layers for the first time. Two separate compartments, which mimic superficial and deep layers, hold blood-lipid mixtures that can be deoxygenated (using yeast) and oxygenated (by bubbling oxygen) independently. Simultaneous NIRS measurements can be performed on the two-layered medium (variable superficial layer thickness, L), the deep (homogeneous), and/or the superficial (homogeneous). In two experiments involving ink, we increased the nominal µ_a in one of two compartments from 0.05 to 0.25 cm^-1, L set to 14.5 mm. In three experiments involving blood (L set to 12, 15, or 17 mm), we used a protocol consisting of six deoxygenation cycles. A state-of-the-art multi-wavelength TD-NIRS system measured simultaneously on the two-layered medium, as well as on the deep compartment for a reference. The new method accurately determined µ_a (and hence StO₂) in both compartments. The method is a significant progress in overcoming the contamination from the superficial layer, which is beneficial for NIRS and fNIRS applications, and may improve the determination of StO₂ in the brain from measurements on the head. The advanced phantom may assist in the ongoing effort towards more realistic standardized performance tests in NIRS tissue oximetry. Data and MATLAB codes used in this study were made publicly available.

Numerical approach to quantify depth-dependent blood flow changes in real-time using the diffusion equation with continuous-wave and time-domain diffuse correlation spectroscopy

Diffuse correlation spectroscopy (DCS) is a non-invasive optical technique that can measure brain perfusion by quantifying temporal intensity fluctuations of multiply scattered light. A primary limitation for accurate quantitation of cerebral blood flow (CBF) is the fact that experimental measurements contain information about both extracerebral scalp blood flow (SBF) as well as CBF. Separating CBF from SBF is typically achieved using multiple source-detector channels when using continuous-wave (CW) light sources, or more recently with use of time-domain (TD) techniques. Analysis methods that account for these partial volume effects are often employed to increase CBF contrast. However, a robust, real-time analysis procedure that can separate and quantify SBF and CBF with both traditional CW and TD-DCS measurements is still needed. Here, we validate a data analysis procedure based on the diffusion equation in layered media capable of quantifying both extra- and cerebral blood flow in the CW and TD. We find that the model can quantify SBF and CBF coefficients with less than 5% error compared to Monte Carlo simulations using a 3-layered brain model in both the CW and TD. The model can accurately fit data at a rate of <10 ms for CW data and <250 ms for TD data when using a least-squares optimizer.

Efficient computation of the steady-state and time-domain solutions of the photon diffusion equation in layered turbid media

Accurate and efficient forward models of photon migration in heterogeneous geometries are important for many applications of light in medicine because many biological tissues exhibit a layered structure of independent optical properties and thickness. However, closed form analytical solutions are not readily available for layered tissue-models, and often are modeled using computationally expensive numerical techniques or theoretical approximations that limit accuracy and real-time analysis. Here, we develop an open-source accurate, efficient, and stable numerical routine to solve the diffusion equation in the steady-state and time-domain for a layered cylinder tissue model with an arbitrary number of layers and specified thickness and optical coefficients. We show that the steady-state ([Formula: see text] ms) and time-domain ([Formula: see text] ms) fluence (for an 8-layer medium) can be calculated with absolute numerical errors approaching machine precision. The numerical implementation increased computation speed by 3 to 4 orders of magnitude compared to previously reported theoretical solutions in layered media. We verify our solutions asymptotically to homogeneous tissue geometries using closed form analytical solutions to assess convergence and numerical accuracy. Approximate solutions to compute the reflected intensity are presented which can decrease the computation time by an additional 2-3 orders of magnitude. We also compare our solutions for 2, 3, and 5 layered media to gold-standard Monte Carlo simulations in layered tissue models of high interest in biomedical optics (e.g. skin/fat/muscle and brain). The presented routine could enable more robust real-time data analysis tools in heterogeneous tissues that are important in many clinical applications such as functional brain imaging and diffuse optical spectroscopy.

Quantification of cerebral blood flow in adults by contrast-enhanced near-infrared spectroscopy: Validation against MRI

The purpose of this study was to assess the accuracy of absolute cerebral blood flow (CBF) measurements obtained by dynamic contrast-enhanced (DCE) near-infrared spectroscopy (NIRS) using indocyanine green as a perfusion contrast agent. For validation, CBF was measured independently using the MRI perfusion method arterial spin labeling (ASL). Data were acquired at two sites and under two flow conditions (normocapnia and hypercapnia). Depth sensitivity was enhanced using time-resolved detection, which was demonstrated in a separate set of experiments using a tourniquet to temporally impede scalp blood flow. A strong correlation between CBF measurements from ASL and DCE-NIRS was observed (slope = 0.99 ± 0.08, y-intercept = -1.7 ± 7.4 mL/100 g/min, and R2 = 0.88). Mean difference between the two techniques was 1.9 mL/100 g/min (95% confidence interval ranged from -15 to 19 mL/100g/min and the mean ASL CBF was 75.4 mL/100 g/min). Error analysis showed that structural information and baseline absorption coefficient were needed for optimal CBF reconstruction with DCE-NIRS. This study demonstrated that DCE-NIRS is sensitive to blood flow in the adult brain and can provide accurate CBF measurements with the appropriate modeling techniques.

Lawrence K, Amendolia O, Quattrone F, Balu R, Kofke WA, Yodh AG. Noninvasive continuous optical monitoring of absolute cerebral blood flow in critically ill adults

We investigate a scheme for noninvasive continuous monitoring of absolute cerebral blood flow (CBF) in adult human patients based on a combination of time-resolved dynamic contrast-enhanced near-infrared spectroscopy (DCE-NIRS) and diffuse correlation spectroscopy (DCS) with semi-infinite head model of photon propogation. Continuous CBF is obtained via calibration of the DCS blood flow index (BFI) with absolute CBF obtained by intermittent intravenous injections of the optical contrast agent indocyanine green. A calibration coefficient ( γ ) for the CBF is thus determined, permitting conversion of DCS BFI to absolute blood flow units at all other times. A study of patients with acute brain injury ( N = 7 ) is carried out to ascertain the stability of γ . The patient-averaged DCS calibration coefficient across multiple monitoring days and multiple patients was determined, and good agreement between the two calibration coefficients measured at different times during single monitoring days was found. The patient-averaged calibration coefficient of 1.24 × 10 9    ( mL / 100    g / min ) / ( cm 2 / s ) was applied to previously measured DCS BFI from similar brain-injured patients; in this case, absolute CBF was underestimated compared with XeCT, an effect we show is primarily due to use of semi-infinite homogeneous models of the head.

Analytical solutions of the radiative transport equation for turbid and fluorescent layered media

Accurate and efficient solutions of the three dimensional radiative transport equation were derived in all domains for the case of layered scattering media. Index mismatched boundary conditions based on Fresnel’s equations were implemented. Arbitrary rotationally symmetric phase functions can be applied to characterize the scattering in the turbid media. Solutions were derived for an obliquely incident beam having arbitrary spatial profiles. The derived solutions were successfully validated with Monte Carlo simulations and partly compared with analytical solutions of the diffusion equation.