Photon Density Waves Research Articles

Measurement of diffuse photon-pairs density wave in a multiple-scattering medium

As a continuation of the previously developed theory of a diffuse photon-pairs density wave (DPPDW) [Appl. Opt.44, 1416-1425 (2005)APOPAI0003-693510.1364/AO.44.001416], this research experimentally studies and verifies the DPPDW theory in a heterogeneous multiple-scattering medium. The DPPDW is generated by collecting the scattered linear polarized photon pairs (LPPPs) in the multiple-scattering medium. Theoretically, the common-path propagation of LPPPs not only provides common phase noise rejection mode but also performs coherence technique via heterodyne detection. In addition, the polarization gating and spatial coherence gating of LPPPs would suppress the severe scattered photon in the multiple-scattering medium. In the experiment, the amplitude and phase wavefronts of DPPDWs, which are distorted by a small object embedded in a homogeneous multiple-scattering medium, are measured in one dimension or two dimensions by scanning the source detector pair. The measured distortion of DPPDW wavefronts are detected precisely and are consistent with the theoretical calculation of DPPDW. It implies an improvement on the detection sensitivity of a small object compared with the conventional diffuse photon density wave (DPDW).

High-frequency limit of photon density waves in turbid media

We propose an iterative scheme to compute quasi-analytically the laser irradiance transmitted through a turbid media illuminated by high-frequency intensity-modulated light. We compare the spatial distribution of the transmitted signal obtained from iterative calculations with that obtained from random walk simulations and from the traditional diffusion theory. In the regime where the diffusion theory breaks down for a highly modulated source, only the ballistic and quasi-ballistic photons survive. This regime could become accessible for interesting imaging applications as the beam width of the AC signal of the transmitted light decreases with an increasing frequency.

Fast intersections on nested tetrahedrons (FINT): An algorithm for adaptive finite element based distributed parameter estimation

A variety of biomedical imaging techniques such as optical and fluorescence tomography, electrical impedance tomography, and ultrasound imaging can be cast as inverse problems, wherein image reconstruction involves the estimation of spatially distributed parameter(s) of the PDE system describing the physics of the imaging process. Finite element discretization of imaged domain with tetrahedral elements is a popular way of solving the forward and inverse imaging problems on complicated geometries. A dual-adaptive mesh-based approach wherein, one mesh is used for solving the forward imaging problem and the other mesh used for iteratively estimating the unknown distributed parameter, can result in high resolution image reconstruction at minimum computation effort, if both the meshes are allowed to adapt independently. Till date, no efficient method has been reported to identify and resolve intersection between tetrahedrons in independently refined or coarsened dual meshes. Herein, we report a fast and robust algorithm to identify and resolve intersection of tetrahedrons within nested dual meshes generated by 8-similar subtetrahedron subdivision scheme. The algorithm exploits finite element weight functions and gives rise to a set of weight functions on each vertex of disjoint tetrahedron pieces that completely cover up the intersection region of two tetrahedrons. The procedure enables fully adaptive tetrahedral finite elements by supporting independent refinement and coarsening of each individual mesh while preserving fast identification and resolution of intersection. The computational efficiency of the algorithm is demonstrated by diffuse photon density wave solutions obtained from a single- and a dual-mesh, and by reconstructing a fluorescent inclusion in simulated phantom from boundary frequency domain fluorescence measurements.

Detection and characterization of breast tumours by time-domain scanning optical mammography

AbstractThe paper gives a short overview of various methods of optical mammography, emphasizing scanning time-domain mammography. The results of a clinical study on time-domain optical mammography are reviewed, comprising 154 patients carrying a total of 102 carcinomas validated by histology. A visibility score attributed to each carcinoma as qualitative measure of tumour detectability indicates acceptable sensitivity but poor specificity for discrimination between malignant and benign lesions. Likewise, a multi-variate statistical analysis yields sensitivity and specificity between 80% and 85% for tumour detection and discrimination with respect to normal (healthy) breast tissue, but values less than 70% for discrimination between malignant and benign breast lesions, being too low to be of clinical relevance. For 87 of the 88 tumours detected retrospectively in both projection optical mammograms, optical properties and tissue parameters were derived based on the diffraction of photon density waves by a spherical inhomogeneity as forward model. Following injection of a bolus of indocyanine green as non-targeted absorbing contrast agent, dynamic contrast-enhanced time-domain optical mammography was carried out on a small number of patients, but no differences in wash-out kinetics of indocyanine green between tumours and healthy breast tissue were observed.

Changes in optical properties of tissue during acute wound healing in an animal model

Changes of optical properties of wound tissue in hairless rats were quantified by diffuse photon density wave methodology at near-infrared frequencies. The diffusion equation for semi-infinite media was used to calculate the absorption and scattering coefficients based on measurements of phase and amplitude with a frequency domain device. There was an increase in the absorption and scattering coefficients and a decrease in blood saturation of the wounds compared with the nonwounded sites. The changes correlated with the healing stage of the wound. The data obtained were supported by immunohistochemical analysis of wound tissue. These results verified now by two independent animal studies could suggest a noninvasive method to detect the progress of wound healing.

Evaluation of higher-order time-domain perturbation theory of photon diffusion on breast-equivalent phantoms and optical mammograms

Time-domain perturbation theory of photon diffusion up to third order was evaluated for its accuracy in deducing optical properties of breast tumors using simulated and physical phantoms and by analyzing 141 projection mammograms of 87 patients with histology-validated tumors that had been recorded by scanning time-domain optical mammography. The slightly compressed breast was modeled as (partially) homogeneous diffusely scattering infinite slab containing a scattering and absorbing spherical heterogeneity representing the tumor. Photon flux densities were calculated from densities of transmitted photons, assuming extended boundary conditions. Explicit formulas are provided for second-order changes in transmitted photon density due to the presence of absorbers or scatterers. The results on phantoms obtained by perturbation theory carried up to third order were compared with measured temporal point spread functions, with numerical finite-element method (FEM) simulations of transmitted photon flux density, with results obtained from the diffraction of diffuse photon density waves, and from Padé approximants. The breakdown of first-, second-, and third-order perturbation theory is discussed for absorbers and a general expression was derived for the convergence of the Born series in this case. Taking tumor optical properties derived by the diffraction model as reference we conclude that estimates of tumor absorption coefficients by perturbation theory agree with reference values within +/-25% in only 65% (first order), 66% (second order), and 77% (third order) of all mammograms analyzed. In the remaining cases tumor absorption is generally underestimated due to the breakdown of perturbation theory. On average the empirical Padé approximants yield tumor absorption coefficients similar to third-order perturbation theory, yet at noticeable lower computational efforts.

Determination of Particle Size and Number Density of Opaque Colloidal Mixtures Using Diffuse Photon Density Waves and Two-Wavelength Light Sources

A diffuse photon density wave (DPDW) propagates as a spherical energy wave in highly scattering media, such as opaque colloidal mixtures. The advantage of using DPDW is that the absorption and reduced scattering coefficients of the opaque colloidal mixtures can be measured without dilution and calibration. We propose a method for the estimation of the mean particle size and number density of the opaque colloidal mixtures using the relationship between estimated values and optical properties. In this study, we first determined the mean particle size and number density of colloidal silica, a single-particle dispersive medium, to confirm the validity of the method and evaluated the accuracy of the measurement. Then, we determined the mean particle size and number density of casein micelles and fat globules in milk, which is regarded as a typical opaque colloidal mixture, using two light sources with different wavelengths.

Signal-to-noise analysis for propagation of laser radiation through a tissue-like medium by diffuse photon-density waves

Biomedical optical imaging in the near-infrared (NIR) region provides the possibility to detect and determine pathological and functional changes in human tissue without the drawback of ionizing radiation. Of special promise is the application of this technology for the detection of joint diseases, such as rheumatoid arthritis (RA). It has been shown that optical changes in the synovial fluid and the vasculature surrounding the joints can be detected with optical methods. Applying optical tomographic methods one should be able to localize and quantify these changes for detection of the onset of RA. The first studies have been limited to continuous wave imaging. However, it is well known that enhanced resolution and better separation between absorption and scattering properties of tissue can be achieved using intensity modulated light sources. Intensity modulation of laser light in the MHz region leads to propagation of so-called diffuse photon density waves (PDW) through the tissue In this study we report on basic experimental results to determine performance and sensitivity of PDW-transillumination of tissue like phantoms. We used a vector network analyzer to generate and analyze intensity modulation from 100 MHz up to 1 GHz via a diode laser and an avalanche photo diode. Scans were performed across phantoms containing a layer with different absorbing and scattering properties bounded by an edge. The thickness of the phantoms was chosen similar to human fingers to gain information for optimization of tomographic imaging of finger joints. We experimentally determined the signal-to-noise ratio (SNR) of the system and compared the results to theoretical predictions. Noise and SNR of amplitude and phase depend on frequency of modulation. While the amplitude SNR decreases with frequency, phase SNR increases to assume a maximum value. We found that the inserted layer can be better characterized using phase information, which becomes more valuable as the source modulation frequency is increased. On the other hand, the sensitivity to perturbations is highest in the amplitude data obtained at lower frequencies. Thus, for tomographic imaging, optimal modulation frequencies should be found depending on the tissue type and nature of tissue inhomogeneities.

Optical Properties of Wounds: Diabetic Versus Healthy Tissue

Diffuse photon density wave (DPDW) methodology at Near Infrared frequencies has been used to calculate absorption and scattering from wounds of healthy and diabetic rats. The diffusion equation for semi-infinite media is being used for calculating the absorption and scattering coefficients based on measurements of phase and amplitude with a frequency domain device. Differences observed during the course of healing in the two populations can be correlated to the delayed healing observed in diabetics. These results are encouraging and further work will focus on the implementation of this device to the clinical setting as a monitoring tool in chronic diabetic wounds.

Experimental results for propagation of diffuse photon-density waves up to 1 GHz in a tissue-like medium containing an absorbing edge

Optical imaging in the near-infrared (NIR) region provides the possibility to detect and determine pathological changes in human tissue without the drawback of ionizing radiation and with little technical and financial effort. Especially in rheumatoid arthritis, imaging by optical tomography to detect early inflammations in joints has the potential to become a supportive tool to common imaging modalities. One way to enhance the resolution and specificity of optical tissue characterization is to use the frequency domain instead of DC intensity measurement. Intensity modulation of a light source leads to propagation of diffuse photon-density waves (PDW) through the tissue. In this study, we report basic experimental results on tissuelike phantoms to determine the optimal parameters for PDW-transillumination of finger joints. We used PDW with modulation frequencies from 100 MHz up to 1 GHz to scan across a tissuelike phantom containing an absorbing plane bounded by an edge. The geometrical extents of the phantoms are similar to human finger joints. We measure the transmitted PDW and show that amplitude and phase behaves at the edge as expected according to theoretical predictions. An increasing modulation frequency leads to increasing slope of the amplitude decay at the edge but decreasing signal-to-noise ratio. Even at 1 GHz, the edge is detectable.

Imaging in turbid media with intensity-modulated lasers

We describe the formation of a narrow beam for intensity-modulated electromagnetic radiation propagating through highly scattering materials. We propose to use this beam to reconstruct images, similar to X-ray back-projection techniques. For sufficiently high modulation frequency, the photon density wave is primarily carried by photons that suffer small or no large-angle scattering, which gives rise to the beam’s narrow divergence. The beam-narrowing concept is supported by large-scale numerical simulations to examine the quality of the imaging.

On the effect of the width of the scattering indicatrix on the dispersion of photon density waves in a strongly scattering medium

The dispersion of photon density waves in strongly scattering media with different widths of the scattering indicatrix is studied by the spherical harmonics method using approximations of various orders (up to the P 7 approximation inclusive). It is shown that, beginning from the P 3 approximation, the reduction in the velocity of photon density waves that is characteristic of the P 1 approximation is eliminated and, independently of the width of the scattering indicatrix in the region of modulation frequencies exceeding 1010 Hz, the velocity of photon density waves asymptotically approaches the speed of light. Our study of the damping of photon density waves has shown that the formula obtained previously for the calculation of the damping coefficient (Imk(μ s ′ , ω)) as a function of the transport scattering coefficient and the velocity is valid at Imk ≤ μs (μs is the light scattering coefficient). The maximum growth in the damping coefficient of photon density waves with a further increase in the frequency is limited by the value of the light scattering coefficient Imk max ≈ μs.

Frequency domain photon migration in theδ−P1approximation: Analysis of ballistic, transport, and diffuse regimes

The standard diffusion approximation (SDA) to the Boltzmann transport equation (BTE) is commonly used to describe radiative transport for biomedical applications of frequency-domain diffuse optical imaging and spectroscopy. Unfortunately, the SDA is unable to provide accurate radiative transport predictions on spatial scales comparable to the transport mean free path and for media in which optical scattering is not dominant over absorption. Here, we develop and demonstrate the use of the delta- P1 approximation to provide improved radiative transport estimates in the frequency domain via the addition of a Dirac delta function to both radiance and phase function approximations. Specifically, we consider photon density wave propagation resulting from the illumination of an infinite turbid medium with an embedded, intensity-modulated, spherical light source. We examine the accuracy of the standard diffusion and delta- P1 approximations relative to Monte Carlo simulations that provide exact solutions to the BTE. This comparison establishes the superior accuracy of the delta- P1 approximation relative to the SDA that is most notable at distances less than 3 transport mean free paths from the source. In addition, we demonstrate that the differences in photon density wave propagation in a highly forward scattering medium (g1=0.95) vs an isotropically scattering medium (g1=0) provides a basis to define three spatial regimes where the light field is dominated by (a) unscattered/ballistic light, (b) minimally scattered light, and (c) diffusely scattered light. We examine the impact of optical properties, source modulation frequency, and numerical aperture of detection on the spatial extent and location of these regimes.

Optical Tomography Imaging Based on Higher Order Born Approximation of Diffuse Photon Density Waves

In this paper, we introduce a tomographical system suitable for imaging absorbers in strongly scattering media using a single light source-detector pair. Their reflective alignment would make it possible to implement both source and detector into a compact scan-head. Suitable probes would be soft tissue, some strongly scattering biological probes, or translucent technical materials, for example. An intensity-modulation technique is utilized to take advantage of the amplitude and delay alterations between the source and the detector. The propagation of the photons is modeled by the Helmholtz equation and then appropriate inverse calculations of several measurements reveal the distribution of the absorption coefficient inside the investigated volume. Therefore a reconstruction algorithm employing a Born approximation of higher order and a Tikhonov-regularization scheme were used to solve the deduced ill-posed and underdetermined system of equations. In this paper, we present results of simulations as well as of measurements.

Examples of the application of optical process and quality sensing (OPQS) to beer brewing and polyurethane foaming processes

Optical methods play an important role in process analytical technologies (PAT). Four examples of optical process and quality sensing (OPQS) are presented, which are based on three important experimental techniques: near-infrared absorption, luminescence quenching, and a novel method, photon density wave (PDW) spectroscopy. These are used to evaluate four process and quality parameters related to beer brewing and polyurethane (PU) foaming processes: the ethanol content and the oxygen (O2) content in beer, the biomass in a bioreactor, and the cellular structures of PU foam produced in a pilot production plant.

Time-domain scanning optical mammography: II. Optical properties and tissue parameters of 87 carcinomas

Within a clinical trial on scanning time-domain optical mammography reported on in a companion publication (part I), craniocaudal and mediolateral projection optical mammograms were recorded from 154 patients, suspected of having breast cancer. Here we report on in vivo optical properties of the subset of 87 histologically validated carcinomas which were visible in optical mammograms recorded at two or three near-infrared wavelengths. Tumour absorption and reduced scattering coefficients were derived from distributions of times of flight of photons recorded at the tumour site employing the model of diffraction of photon density waves by a spherical inhomogeneity, located in an otherwise homogeneous tissue slab. Effective tumour radii, taken from pathology, and tumour location along the compression direction, deduced from off-axis optical scans of the tumour region, were included in the analysis as prior knowledge, if available. On average, tumour absorption coefficients exceeded those of surrounding healthy breast tissue by a factor of about 2.5 (670 nm), whereas tumour reduced scattering coefficients were larger by about 20% (670 nm). From absorption coefficients at 670 nm and 785 nm total haemoglobin concentration and blood oxygen saturation were deduced for tumours and surrounding healthy breast tissue. Apart from a few outliers total haemoglobin concentration was observed to be systematically larger in tumours compared to healthy breast tissue. In contrast, blood oxygen saturation was found to be a poor discriminator for tumours and healthy breast tissue; both median values of blood oxygen saturation are the same within their statistical uncertainties. However, the ratio of total haemoglobin concentration over blood oxygen saturation further improves discrimination between tumours and healthy breast tissue. For 29 tumours detected in optical mammograms recorded at three wavelengths (670 nm, 785 nm, 843 nm or 884 nm), scatter power was derived from transport scattering coefficients. Scatter power of tumours tends to be larger than that of surrounding healthy breast tissue, yet the 95% confidence intervals of both medians overlap.

Measurement of the coagulation dynamics of bovine liver using the modified microscopic Beer–Lambert law

During heating, the optical properties of biological tissues change with the coagulation state. In this study, we propose a technique, which uses these changes to monitor the coagulation process during laser-induced interstitial thermotherapy (LITT). Untreated and coagulated (water bath, temperatures between 35 degrees C and 90 degrees C for 20 minutes.) samples of bovine liver tissue were examined using a Nd:YAG (lambda = 1064 nm) frequency-domain reflectance spectrometer. We determined the time integrated intensities (I(DC)) and the phase shifts (Phi) of the photon density waves after migration through the tissue. From these measured quantities, the time of flight (TOF) of the photons and the absorption coefficients of the samples were derived using the modified microscopic Beer-Lambert law. The absorption coefficients of the liver samples decreased significantly with the temperature in the range between 50 degrees C and 70 degrees C. At the same time, the TOF of the investigated photos was found increased indicating an increased scattering. The coagulation dynamics could be well described using the Arrhenius formalism with the activation energy of 106 kJ/mol and the frequency factor of 1.59 x 10(13)/second. Frequency-domain reflectance spectroscopy in combination with the modified microscopic Beer-Lambert (MBL) is suitable to measure heat induced changes in the absorption and scattering properties of bovine liver in vitro. The technique may be used to monitor the coagulation dynamics during local thermo-coagulation in vivo.

Locating inhomogeneities in tissue by using the most probable diffuse path of light

Characterization of human tissue using near-IR (NIR) light is becoming increasingly popular. The light signal transmitted from the tissue contains information concerning inhomogeneities in tissue, such as size, position, and pathological states (benign or malignant). We discuss the most probable diffuse path (MPDP) related to frequency-domain diffuse photon density waves (DPDWs) propagating inside turbid media. We find that for a medium of finite size, the existence of boundaries between tissue and nonscattering media would have considerable impact on the path shape. It is also demonstrated that such paths can be used to obtain higher accuracy in localizing absorbers embedded in a homogeneous background. Based on the proposed MPDP, a new method for 3-D localization of heterogeneities in turbid media is proposed, which is validated by experiments using Intralipid and pork fat. The experiments are performed with an NIR breast cancer detection system designed and assembled in our lab, using 780-nm NIR light. In Intralipid, when the size of a single absorber is less than 1 cm, the localization error is about 2 mm. The results from pork fat are also acceptable.

Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography

Using a dual-wavelength (670 nm, 785 nm) time-domain scanning instrument we have recorded optical mammograms of 93 patients suspected of having breast cancer which was subsequently assessed histologically. Among 65 histologically confirmed carcinomas, 54 were detectable in at least one of two optical mammograms recorded of each tumour-bearing breast in craniocaudal and mediolateral projection. Optical mammograms were based on photon counts in selected time windows of measured distributions of times of flight of photons. Optical properties of 50 carcinomas investigated at both wavelengths were derived by modelling the breast as partially homogeneous infinite slab with an embedded spherical inhomogeneity representing the tumour and by calculating the diffraction of photon density waves. In selected cases, additional information about the location of the tumour along the compression direction was used that was obtained from scans at selected offsets between source and detector optical fibres. A correlation plot of haemoglobin concentration and blood oxygen saturation of tumours and healthy tissue shows good separation between both kinds of tissue. The majority of carcinomas exhibited increased total haemoglobin concentration compared to healthy tissue.

Green functions for diffuse photon-density waves generated by a line source in two nonabsorbing turbid media in contact

Diffuse photon-density waves generated by an instantaneous line source that is parallel to the interface between two semi-infinite turbid media are studied by use of the diffusion approximation. For two nonabsorbing media the Green functions for diffuse light are obtained based on the Green functions for temperature fields that were derived with the Cagniard-de Hoop method. The boundary conditions for diffuse light take into account the discontinuity in the specific intensity at the interface between two media with different refractive indices. The results of the calculations of the specific intensities and the gradient lines for different sets of parameters are presented.