Optical spectroscopy of turbid media: time-domain measurements and accelerated Monte Carlo modelling

Optical spectroscopy is a versatile and powerful tool to probe translucent materials. In this work, the focus is on characterization of strongly scattering (turbid) materials by means of time-of-flight spectroscopy (TOFS). Instrumentation and modelling aspects of TOFS were investigated and improved, enabling significantly more accurate spectroscopic measurements. It was shown that the commonly used diffusion theory fails to accurately describe time-domain light propagation in e.g. tissue. A fully scalable Monte Carlo (MC) scheme (WMC) was developed, enabling MC to replace diffusion models in TOFS data evaluation. Consequently, the accuracy and capabilities of TOFS were significantly improved. Graphics processing units (GPUs) were introduced for acceleration of MC simulations in general, resulting in three orders of magnitude speedup. It was shown that proper utilization of the capabilities of modern GPUs allow similar performance, even for more complex problems. TOFS in combination with WMC was used in in vivo interstitial spectroscopy of the human prostate, demonstrating the need for better modelling in many clinical applications. To aid future interstitial in vivo measurements, a single-fibre TOFS system was developed and demonstrated in phantom experiments. Turning to investigations of pharmaceutical samples, a time-of-flight spectrometer, covering the 650-1400 nm spectral range, was developed, enabling TOFS for vibrational spectroscopy of solids. In spatially resolved TOFS measurements, compaction induced anisotropic light diffusion was observed. This is of great importance for the application of model-based optical spectroscopic tech- niques and may, in addition, provide important information about the sample microstructure. Furthermore, TOFS was used together with laser-based gas sensing to probe porous solids. Although a need for better models was revealed, excellent correlation between optical and actual porosity was demonstrated

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 novel 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 (SOS) 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

Two-step verification method for Monte Carlo codes in biomedical optics applications

Significance: Code verification is an unavoidable step prior to using a Monte Carlo (MC) code. Indeed, in biomedical optics, a widespread verification procedure for MC codes is still missing. Analytical benchmarks that can be easily used for the verification of different MC routines offer an important resource.Aim: We aim to provide a two-step verification procedure for MC codes enabling the two main tasks of an MC simulator: (1) the generation of photons' trajectories and (2) the intersections of trajectories with boundaries separating the regions with different optical properties. The proposed method is purely based on elementary analytical benchmarks, therefore, the correctness of an MC code can be assessed with a one-sample t-test.Approach: The two-step verification is based on the following two analytical benchmarks: (1) the exact analytical formulas for the statistical moments of the spatial coordinates where the scattering events occur in an infinite medium and (2) the exact invariant solutions of the radiative transfer equation for radiance, fluence rate, and mean path length in media subjected to a Lambertian illumination.Results: We carried out a wide set of comparisons between MC results and the two analytical benchmarks for a wide range of optical properties (from non-scattering to highly scattering media, with different types of scattering functions) in an infinite non-absorbing medium (step 1) and in a non-absorbing slab (step 2). The deviations between MC results and exact analytical values are usually within two standard errors (i.e., t-tests not rejected at a 5% level of significance). The comparisons show that the accuracy of the verification increases with the number of simulated trajectories so that, in principle, an arbitrary accuracy can be obtained.Conclusions: Given the simplicity of the verification method proposed, we envision that it can be widely used in the field of biomedical optics.</p

New Horizons in Time-Domain Diffuse Optical Spectroscopy and Imaging

Jöbsis was the first to describe the in vivo application of near-infrared spectroscopy (NIRS), also called diffuse optical spectroscopy (DOS). NIRS was originally designed for the clinical monitoring of tissue oxygenation, and today it has also become a useful tool for neuroimaging studies (functional near-infrared spectroscopy, fNIRS). However, difficulties in the selective and quantitative measurements of tissue hemoglobin (Hb), which have been central in the NIRS field for over 40 years, remain to be solved. To overcome these problems, time-domain (TD) and frequency-domain (FD) measurements have been tried. Presently, a wide range of NIRS instruments are available, including commonly available commercial instruments for continuous wave (CW) measurements, based on the modified Beer–Lambert law (steady-state domain measurements). Among these measurements, the TD measurement is the most promising approach, although compared with CW and FD measurements, TD measurements are less common, due to the need for large and expensive instruments with poor temporal resolution and limited dynamic range. However, thanks to technological developments, TD measurements are increasingly being used in research, and also in various clinical settings. This Special Issue highlights issues at the cutting edge of TD DOS and diffuse optical tomography (DOT). It covers all aspects related to TD measurements, including advances in hardware, methodology, the theory of light propagation, and clinical applications

Stochastic processes for anomalous diffusion

161 p.Con difusión anómala se hace referencia a procesos de difusión en los cuales el desplazamiento cuadrático medio (MSD) no es una función lineal de la variable tiempo (lo que se conoce como difusión normal). Cuando la relación es más rápida que lineal, se le llama superdifusión, y cuando es más lenta, subdifusión. Físicamente, el MSD es una medida de las desviación de la posición de las partículas con el tiempo. Se puede imaginar como la cantidad de espacio que las partículas han explorado en el sistema.La difusión anómala aparece constantemente en la naturaleza y por ello los científicos han desarrollado diferentes modelos que pueden reproducirla efectivamente. Sin embargo, la física subyacente de una gran cantidad de experimentos aun no se comprende correctamente. Este es el caso del movimiento de las moléculas de ARN mensajero dentro de bacterias E. coli, donde los más importantes procesos estocásticos fallan al intentar explicar todas sus características al mismo tiempo. Por ejemplo, el caminante aleatorio con tiempo continuo (CTRW) puede explicar la falta de ergodicidad pero no la variación-p. Por otro lado, el movimiento Browniano fraccionario (fBm) puede explicar su variación-p pero se trata de un proceso ergódico. Hemos analizado una clase de procesos estocásticos conocida como movimiento Browniano gris generalizado (ggBm) que además de reproducir difusión anómala, no es ergódico y tiene una variación-p como la del fBm, siendo por tanto un buen candidato para explicar el experimento citado anteriormente. Además, hemos incluído un factor con una dependencia temporal explícita para que sea un proceso no estacionario, característica que se suele denominar envejecimiento. De forma coincidente con los resultados computacionales, hemos podido encontrar expresiones matemáticas para muchos observables incluyendo el MSD promediado colectivamente y temporalmente, el parámetro de Rotura de la Ergodicidad, la función densidad de probabilidad en un punto y en un tiempo y la variación-p.Por otro lado, muchos experimentos muestran una transición característica de un régimen de difusión anómala a otro de difusión normal. Por ejemplo, esto ocurre en sistemas viscoelásticos como el movimiento de moléculas de lípidos en membranas bicapa de lípidos. La segunda parte de esta tesis está dedicada al estudio de procesos estocásticos que mediante una truncamiento de la función de autocorrelación del ruido o fuerza estocástica en la ecuación de Langevin sobredimensionada o la ecuación de Langevin fraccionaria sobredimensionada, se puede conseguir este tipo de transición. Esto ocurre cuando el truncamiento se hace de forma exponencial o mediante una ley de potencias suficientemente fuerte. Si la ley de potencias es débil, se obtienen transiciones de un régimen de rápida superdifusión a otra más lenta, y de un régimen de subdifusión lenta a otra más rápida, respectivamente. En esta parte también se consideran otros procesos en los que el truncamiento se realiza directamente sobre la definición del fBm original de Mandelbrot y Van Ness. Se conoce como movimiento Browniano fraccionario templado (tfBm) y, sorprendentemente, no tiene las mismas propiedades que los anteriores modelos. En su lugar, a tiempos largos presenta localización como el proceso de Ornstein-Uhlenbeck. Hemos comparado ambos procesos. Cuando la derivada del tfBm es usada en ecuación de Langevin fraccionaria, se obtiene difusión balística para tiempos largos.En la última parte estamos interesados en los últimos avances en teoría neutral espacial del campo de la Ecología, cuyos problemas resultan familiares a los expertos en Física Estadística. De hecho, los ecosistemas reproducen una organización espacial compleja que los ecólogos han intentando caracterizar observando los diferentes patrones de biodiversidad a distintas escalas espaciales. Relacionar estas medidas con las causas que las originan es probablemente el problema central de la Ecología. La teoría neutral de la Ecología, que subraya el papel de las fluctuaciones demográficas estocásticas y rechaza los efectos deterministas procedentes de diferencias adaptativas, ha predicho los patrones empíricos en comunidades de especies que compiten entre si. Hemos estudiado las leyes de escalado que surgen en la dimensión crítica (2D) de los modelos espaciales neutrales, los cuales no se comprenden del todo a día de hoy. Acabamos discutiendo algunos modelos con características no neutrales. bcam: basque center for applied mathematicsbcam: basque center for applied mathematic

Stochastic processes for anomalous diffusion

Anomalous diffusion is a diffusion process which Mean Square Displacement (MSD) is not a linear funtion of time, what is known as normal diffusion. When the relation is faster than linear, it is called superdiffusion and when it is slower, subdiffusion. Physically, the MSD is a measure of the deviation of the position of the particles over time. It can be thought as the amount of space the particles have explored in the system. On the one hand, anomalous diffusion has been shown to appear extensively in nature and, consecuently, scientists have developed several and different models that can effectively reproduce it. However, the underlying physics of a plethora of experiments is still not well-understood. This is the case of the motion of mRNA molecules inside living E. coli cells, where stochastic processes as the continuous-time random walk can explain non-ergodicity but an alternative like the fractional Brownian motion is needed to reproduce p-variation. We find that a family of stochastic processes known as generalised grey Brownian motion can fit correctly that observation. In addition, we also use an explicit time-dependent factor to model non-stationarity, usually referred to aging in the specialized literature. In agreement with our computational results, we were able to obtain analytical expressions for a list of observables too, including temporal and ensemble-average MSD, Ergodicity Breaking parameter, one-point one-time probability density function and p-variation. On the other hand, many experiments show a characteristic crossover from anomalous to normal diffusion. For example, it happens in viscoelastic systems such as the motion of lipid molecules in lipid bilayer membranes. We studied the motion of particles driven by tempered fractional Gaussian noise which power-law correlations present a cutoff at some mesoscopic time scale. Deriving analytical expressions of the MSD for the overdamped Langevin equation and the fractional Langevin equation, we find that when the truncation is strongly done, the mentioned crossover appears. When the truncation is done by a weak power-law truncation, we also got a different crossover behaviour from faster to slower superdiffusion and from slower to faster subdiffusion. We were also interested in another process defined through a tempering directly done in the fractional Brownion motion definition. It is known as tempered fractional Brownian motion and, surprisingly, it does not arrive to the same behaviours than our models. Instead, at long times it exhibits localization as Ornstein-Uhlenbeck. We compare both of them. When the derivative of this tempered fractional Brownian motion is used in the fractional Langevin equation, it leads to ballistic diffusion at long times. In the last part, we were interested in the state of the art in spatial neutral theory of Ecology, which problems sound familiar to experts in Statistical Physics. In fact, ecosystems display a complex spatial organization that ecologists have tried to characterize by observing patterns of biodiversity at different spatial scales. Linking those measures with the causes that originate them is arguably the central problem in Ecology. Ecological neutral theory, which underscores the role of stochastic demographic fluctuations and neglects deterministic effects stemming from fitness differences, has predicted the empirical patterns in communities of competing species. We study non-trivial scaling laws arising at the critical dimension (2D) of spatial neutral models, which are poorly understood. We conclude by discussing models that support non-neutral features

Optical Cerebral Blood Flow Monitoring of Mice to Men

This thesis describes cerebral hemodynamic monitoring with the optical techniques of diffuse optical spectroscopy (DOS) and diffuse correlation spectroscopy (DCS). DOS and DCS both employ near-infrared light to investigate tissue physiology millimeters to centimeters below the tissue surface. DOS is a static technique that analyzes multispectral tissue-scattered light intensity signals with a photon diffusion approach (Chapter 2) or a Modified Beer-Lambert law approach (Chapter 3) to derive tissue oxy- and deoxy-hemoglobin concentrations, which are in turn used to compute tissue oxygen saturation and blood volume (Section 2.13). DCS is a qualitatively different dynamic technique that analyzes rapid temporal fluctuations in tissue-scattered light with a correlation diffusion approach to derive tissue blood flow (Chapter 4). Further, in combination these measurements of blood flow and blood oxygenation provide access to tissue oxygen metabolism (Section 7.6). The new contributions of my thesis to the diffuse optics field are a novel analysis technique for the DCS signal (Chapter 5), and a novel approach for separating cerebral hemodynamic signals from extra-cerebral artifacts (Chapter 6). The DCS analysis technique extends the Modified Beer-Lambert approach for DOS to the DCS measurement. This new technique has some useful advantages compared to the correlation diffusion approach. It facilitates real-time flow monitoring in complex tissue geometries, provides a novel route for increasing DCS measurement speed, and can be used to probe tissues wherein light transport is non-diffusive (Chapter 5). It also can be used to filter signals from superficial tissues. For separation of cerebral hemodynamic signals from extra-cerebral artifacts, the Modified Beer-Lambert approach is employed in a pressure modulation scheme, which determines subject-specific contributions of extra-cerebral and cerebral tissues to the DCS/DOS signals by utilizing probe pressure modulation to induce variations in extra-cerebral hemodynamics while cerebral hemodynamics remain constant (Chapter 6). In another novel contribution, I used optical techniques to characterize neurovascular coupling at several levels of cerebral ischemia in a rat model (Chapter 7). Neurovascular coupling refers to the relationship between increased blood flow and oxygen metabolism and increased neuronal activity in the brain. In the rat, localized neuronal activity was increased from functional forepaw stimulation. Under normal flow levels, I (and others) observed that the increase in cerebral blood flow (surrogate for oxygen delivery) from forepaw stimulation exceeded the increase in cerebral oxygen metabolism by about a factor of 2. My measurements indicate that this mismatch between oxygen delivery and consumption are more balanced during ischemia (Chapter 7). In Chapters 2 and 3, I review the underlying theory for the photon diffusion model and the Modified Beer-Lambert law for DOS analysis. I also review the correlation diffusion approach for analyzing DCS signals in Chapter 4. My hope is that readers new to the field will find these background chapters helpful

Diffusive transport: theory and application

In this thesis we investigate the behaviour of diffusing particles in a variety of scenarios. We are primarily interested in the case of molecules diffusing inside a cell in the context of biological processes where the mechanism by which a cell responds to an event occurring on its surface may involve the transport of molecular complexes from the cellular surface to the nucleus, and the transport of synthesised molecules from the nuclear surface to the cellular surface. We find the Green's functions for diffusions in two and three dimensions, respectively, on a domain bounded by non-concentric surfaces, one absorbing and one reflecting. Exact expressions are also found for mean hitting times and hitting densities. Our motivation is diffusive transport from a nuclear surface, to a cellular surface and back. Hence, we consider cases where the initial condition is uniformly distributed on the nuclear or cellular surface, and where the hitting density of the outward leg is the density of initial conditions for the return leg. Mean times are calculated by integrating the Green's functions over the domain. Additionally, we create a mathematical model for a specific type of assay experiments where Coxiella burnetii bacteria are placed inside a well and are allowed to be phagocytosed by a monolayer of monocytes on the bottom of the well. We obtain an expression for the intracellular bacterial load at any point during the experiment