Fluorescence molecular lifetime tomography based on asymptotic source function approximation: prospects for solving the problem

The use of light to explore biology can be traced to the first observations of tissue made with early microscopes in the mid-seventeenth century, and has today evolved into the discipline which we now know as biophotonics. This field encompasses a diverse range of activities, each of which shares the common theme of exploiting the interaction of light with biological material. With the rapid advancement of ultrafast optical technologies over the last few decades, ultrafast lasers have increasingly found applications in biophotonics, to the extent that the distinctive new field of ultrafast biophotonics has now emerged, where robust turnkey ultrafast laser systems are facilitating cutting-edge studies in the life sciences to take place in everyday laboratories. The broad spectral bandwidths, precision timing resolution, low coherence and high peak powers of ultrafast optical pulses provide unique opportunities for imaging and manipulating biological systems. Time-resolved studies of bio-molecular dynamics exploit the short pulse durations from such lasers, while other applications such as optical coherence tomography benefit from the broad optical bandwidths possible by using super-continuum generation and additionally allowing for high speed imaging with speeds as high as 47 000 scans per second. Continuing progress in laser-system technology is accelerating the adoption of ultrafast techniques across the life sciences, both in research laboratories and in clinical applications, such as laser-assisted in situ keratomileusis (LASIK) eye surgery. Revolutionizing the field of optical microscopy, two-photon excitation fluorescence (TPEF) microscopy has enabled higher spatial resolution with improved depth penetration into biological specimens. Advantages of this nonlinear optical process include: reduced photo-interactions, allowing for extensive imaging time periods; simultaneously exciting multiple fluorescent molecules with only one excitation wavelength; and reduced chromatic aberration effects. These extensive advantages have led to further exploration of nonlinear processes including second-harmonic generation (SHG) microscopy and third-harmonic generation (THG) microscopy. Second-harmonic generation has provided biologists with an extremely powerful tool for generating contrast in biological imaging, with the additional benefit of non-invasive three-dimensional imaging. The recent popularity of THG microscopy is largely due to the fact that three-dimensional imaging is achievable without the need for any labels, but rather relying on the intrinsic properties of the biological specimen itself. This optical nonlinear technique has attracted much attention recently from the biological community due to its non-invasive capabilities. Users of ultrafast lasers in the biological and medical fields are becoming a fast-growing community, employing pulse-shaping microscopy,resolution-enhancing microscopy techniques, linear and nonlinear micro-spectroscopy, functional deep-tissue imaging, optical coherence tomography, nonlinear fluorescence microscopy, molecular imaging and control, harmonic microscopy and femtosecond lifetime imaging, for cutting-edge research concerning the interaction of light with biological dynamics. The adaptability of ultrafast lasers to interact with a large array of materials through nonlinear excitation has enabled precise control of laser fluence allowing for highly localized material interactions, permitting micro-structured fabricated surfaces. The resultant multi-dimensional fabricated micro-structures are capable of replicating and/or manipulating microenvironments for controlled cell biology. In this special issue of Journal of Optics readers have a chance to view a collection of new contributions to the growing research field of ultrafast biophotonics. They are presented with recent advances in ultrafast technology applied to biological and medical investigations, where topics include advances in the visualization and identification of photo-reaction dynamics of biological functions under relevant physiological conditions, theoretically proposed imaging designs for obtaining super-resolved optical sectioned images in single exposures and fabricated micro-structured surfaces for biological micro-environments. We hope the collection will stimulate innovative new research in this growing field by showcasing new techniques for the visualization and manipulation of complex biological systems using linear and and nonlinear optical processes. Professor Min Gu would like to acknowledge Dr Betty Kouskousis for her contribution and support towards this editorial.

This articles presents several new and simple closed-form asymptotic approximations for the average symbol error rate (ASER) performance measure of digital communication systems (with/without diversity transceivers) impaired by additive white Gaussian noise (AWGN) and multipath fading. In particular, a new class of asymptotic approximations for the ASER is derived by utilizing a normalized asymptotic probability density function (PDF) of signal-to-noise ratio (SNR). The resulting expressions are considerably better/tighter over a wider range of mean SNRs when compared to the existing high-SNR approximations although both techniques need only the first non-zero term of the Maclaurin (if exists) or the Poincare series expansion of the receiver output SNR. We also investigate the utility/efficacy of the Welch-Satterthwaite and Moschopoulos approximations (based on either a moments matching technique or using our proposed normalized asymptotic PDF of SNR that resembles a Gamma PDF) for yielding accurate predictions of the ASER in the low-SNR regime in a myriad of fading environments. Simple closed-form approximations for the ergodic channel capacities of different types of fading channels with/without diversity reception are also derived.

The paper is devoted to an original method of time-resolved fluorescence molecular tomog-raphy based on asymptotic approximation to the fluorescence source function. It analyzes re-sults the authors have obtained during last 4 years in their testing the method in numerical and physical experiments. The method is inferred to be quite promising and directions of fur-ther research for its verification as a sub-millimeter resolution method are outlined. As a modernization of the method, the authors propose an iterative approach where the fluoro-phore absorption coefficient and fluorescence lifetime are reconstructed successively using a posteriori knowledge on the object from previous iterations.

We study asymptotic approximations of some special functions arising in atomic physics. These ones are closely related to Anger's functions. Our major tool is the method of the stationary phase, the functions under consideration being examples of oscillatory integrals that depend on a parameter $\rho,$ such as $$F(x,\rho):=\int g(\rho, \theta) e^{ix\psi(\theta)} d\theta,$$ We give their behavior when $x$ tends to $\infty$, depending on assumptions on $\rho$. One has to take into account the influence of the two variables. This asks for adaptations of the method of the stationary phase.

Asymptotic approximation for the probability density function of an arbitrary sequence of random variables

BEM solution of Poisson's equation in two dimensions with polynomial interpolation of the source function

The novel Time-Domain Mesoscopic Fluorescence Molecular Tomography (TD-MFMT) imaging modality is aimed to resolve intra-tumoral heterogeneity and monitoring target engagement at the mesoscopic regime; however, its spatial resolution is limited by the scan strategy employed for imaging. Imaging in the Short-Wave Infrared (SWIR) can improve resolution due to reduced scattering effect. We therefore implement data-fusion of low-resolution fluorescence lifetime images with high-resolution SWIR intensity images via deep learning (DL). This approach was validated through phantom study where results show resolution enhancement of fluorescence lifetime imaging in scattering media, demonstrating potential for expansion to preclinical studies.

Empirical Discriminability of Two Models for Stochastic Relationship Between Additive Components of Response Time

Non-Ramanujan Asymptotic Approximation for Zero-Balanced Hypergeometric Functions

We study the representation, approximation, and compression of functions in M dimensions that consist of constant or smooth regions separated by smooth (M-1)-dimensional discontinuities. Examples include images containing edges, video sequences of moving objects, and seismic data containing geological horizons. For both function classes, we derive the optimal asymptotic approximation and compression rates based on Kolmogorov metric entropy. For piecewise constant functions, we develop a multiresolution predictive coder that achieves the optimal rate-distortion performance; for piecewise smooth functions, our coder has near-optimal rate-distortion performance. Our coder for piecewise constant functions employs surflets, a new multiscale geometric tiling consisting of M-dimensional piecewise constant atoms containing polynomial discontinuities. Our coder for piecewise smooth functions uses surfprints, which wed surflets to wavelets for piecewise smooth approximation. Both of these schemes achieve the optimal asymptotic approximation performance. Key features of our algorithms are that they carefully control the potential growth in surflet parameters at higher smoothness and do not require explicit estimation of the discontinuity. We also extend our results to the corresponding discrete function spaces for sampled data. We provide asymptotic performance results for both discrete function spaces and relate this asymptotic performance to the sampling rate and smoothness orders of the underlying functions and discontinuities. For approximation of discrete data, we propose a new scale-adaptive dictionary that contains few elements at coarse and fine scales, but many elements at medium scales. Simulation results on synthetic signals provide a comparison between surflet-based coders and previously studied approximation schemes based on wedgelets and wavelets.

A novel class of special functions for electromagnetics is presented. Formed by the incomplete Hankel and modified Bessel functions, this class allows solving electromagnetics problems concerning truncated cylindrical structures. The differential and recurrence equations of these functions feature additional terms with respect to the classical theory of the Hankel and Bessel functions. The general properties, the most important analytical characteristics, and the large argument asymptotic approximations of the incomplete functions are derived using the steepest descent path (SDP) technique, showing that each special function splits into two terms. The first one has a discontinuous character and is linked to the saddle-point(s) contribution(s), while the second one, arising from the integral end-point contribution(s), compensates exactly the said discontinuity. In the solution of electromagnetic problems, the first term describes the geometrical optics (GO) field, the diffracted field being described by the second one. The general theory is employed to find the closed form analytical solution of the field radiated from a uniform line current source. Using the properties of the incomplete Hankel functions, it is demonstrated that this source excites cylindrical fields having optical character. Finally, the shape of the spatial regions where the GO solution cannot be applied is determined and discussed in details.

The fluctuation theorem for entropy production is a remarkable symmetry of the distribution of produced entropy that holds universally in non-equilibrium steady states with Markovian dynamics. However, in systems with slow degrees of freedom that are hidden from the observer, it is not possible to infer the amount of produced entropy exactly. Previous work suggested that a relation similar to the fluctuation theorem may hold at least approximately for such systems if one considers an apparent entropy production. By extending the notion of apparent entropy production to discrete bipartite systems, we investigate which criteria have to be met for such a modified fluctuation theorem to hold in the large deviation limit. We use asymptotic approximations of the large deviation function to show that the probabilities of extreme events of apparent entropy production always obey a modified fluctuation theorem and, moreover, that it is possible to infer otherwise hidden properties. For the paradigmatic case of two coupled colloidal particles on rings the rate function of the apparent entropy production is calculated to illustrate this asymptotic behavior and to show that the modified fluctuation theorem observed experimentally for short observation times does not persist in the long time limit.

Computing the Kummer function U(a, b, z) for small values of the arguments

In this paper, we review a retrial queue with MMPP input and two-way communication. Incoming requests arriving at the server and finding it busy join the source of retrial calls and try to enter the server again after some exponentially distributed time. While idle, the server makes outgoing calls and serves them with another delay parameter. MMPP (Markov Modulated Poisson Process) is an input process in which control is driven by a continuous Markov chain. Changing its state entails a change in the intensity of the input process. For this model, we present an asymptotic approximation of the two-dimensional characteristic function under the condition of a long delay of requests in the source of retrial calls. For this approximation, we carried out a numerical experiment, where asymptotic results were compared to computations obtained via simulation.KeywordsOutput processRetrial queueTwo-way communicationAsymptotic analysis methodSimulationMarkov modulated poisson process

We assume that a charged quantum particle moves in a space of dimension d = 2, 3,... and is scattered by a fixed Coulomb center. We derive and study expansions of the wave function and all radial functions of such a particle in integer powers of the wave number and in Bessel functions of a real order. We prove that finite sums of such expansions are asymptotic approximations of the wave functions in the low-energy limit.