Computational evaluation of light propagation in cylindrical bioreactors for optogenetic mammalian cell cultures

Light-inducible regulation of cellular pathways and gene circuits in mammalian cells is a new frontier in mammalian genetic engineering. Optogenetic mammalian cell cultures, which are light-sensitive engineered cells, utilize light to regulate gene expression and protein activity. As a low-cost, tunable, and reversible input, light is highly adept at spatiotemporal and orthogonal regulation of cellular behavior. However, light is absorbed and scattered as it travels through media and cells, and the applicability of optogenetics in larger mammalian bioreactors has not been determined. In this work, we computationally explore the size limit to which optogenetics can be applied in cylindrical bioreactors at relevant height-to-diameter ratios. We model the propagation of light using the radiative transfer equation and consider changes in reactor volume, absorption coefficient, scattering coefficient, and scattering anisotropy. We observe sufficient light penetration for activation in simulated bioreactors with sizes of up to 80,000 L at maximal cell densities. We performed supporting experiments and found that significant attenuation occurs at the boundaries of the system, but the relative change in intensity distribution within the reactor was consistent with simulation results. We conclude that optogenetics can be applied to bioreactors at an industrial scale and may be a valuable tool for specific biomanufacturing applications.

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.

A Comparative Analysis of Retrieval Algorithms of Land Surface Temperature from Landsat-8 Data: A Case Study of Shanghai, China

In the trend of global warming and urbanization, frequent extreme weather has a severe impact on the lives of citizens. Land Surface Temperature (LST) is an essential climate variable and a vital parameter for land surface processes at local and global scales. Retrieving LST from global, regional, and city-scale thermal infrared remote sensing data has unparalleled advantages and is one of the most common methods used to study urban heat island effects. Different algorithms have been developed for retrieving LST using satellite imagery, such as the Radiative Transfer Equation (RTE), Mono-Window Algorithm (MWA), Split-Window Algorithm (SWA), and Single-Channel Algorithm (SCA). A case study was performed in Shanghai to evaluate these existing algorithms in the retrieval of LST from Landsat-8 images. To evaluate the estimated LST accurately, measured data from meteorological stations and the MOD11A2 product were used for validation. The results showed that the four algorithms could achieve good results in retrieving LST, and the LST retrieval results were generally consistent within a spatial scale. SWA is more suitable for retrieving LST in Shanghai during the summer, a season when the temperature and the humidity are both very high in Shanghai. Highest retrieval accuracy could be seen in cultivated land, vegetation, wetland, and water body. SWA was more sensitive to the error caused by land surface emissivity (LSE). In low temperature and a dry winter, RTE, SWA, and SCA are relatively more reliable. Both RTE and SCA were sensitive to the error caused by atmospheric water vapor content. These results can provide a reasonable reference for the selection of LST retrieval algorithms for different periods in Shanghai.

Investigating the Effect of Lockdown During COVID-19 on Land Surface Temperature: Study of Dehradun City, India

Urban environment imposes challenges due to its dynamics and thermodynamic characteristics of the built environment. The present study aims to study the effect of lockdown during COVID-19 on the spatio-temporal land surface temperature (LST) patterns in Dehradun city. The TIRS sensor data of 14 April 2020 (post-lockdown), 28 April 2019, 25 April 2018 and 08 May 2017 were downloaded, and LST was retrieved using radiative transfer equation. The wardwise change in LST, urban hot spots and thermal comfort was studied as a function of built-up density. It was observed that there was an overall decrease in LST values in Dehradun city in post-COVID lockdown period. Wards with high built-up density had minimum decrease in LST; on the contrary, wards with large proportion of open spaces and having low, medium built-up density had the maximum decrease in LST. Hot spot analysis was carried out using Getis Ord GI* statistic, and the level of thermal comfort was found using the urban thermal field variance index. It was observed that there was an increase in number of hot spots accompanied by a decrease in thermal comfort level post-lockdown. The methodology proposed in the present study can be applied to other Indian cities which exhibit similar growth patterns and will provide a tool for rational decision making.

Three-dimensional frequency-domain optical anisotropy imaging of biological tissues with near-infrared light

PURPOSE

Near-infrared optical imaging aims to reconstruct the absorption μ a and scattering μ s coefficients in order to detect tumors at early stage. However, the reconstructions have only been limited to μ a and μ s due to theoretical and computational limitations. The authors propose an efficient method of the reconstruction, in three-dimensional geometries, of the anisotropy factor g of the Henyey-Greenstein phase function as a new optical imaging biomarker.

METHODS

The light propagation in biological tissues is accurately modeled by the radiative transfer equation (RTE) in the frequency-domain. The reconstruction algorithm is based on a gradient-based updating scheme. The adjoint method is used to efficiently compute the gradient of the objective function which represents the discrepancy between simulated and measured boundary data. A parallel implementation is carried out to reduce the computational time.

RESULTS

We show that by illuminating only one surface of a tissue-like phantom, the algorithm is able to accurately reconstruct optical values and different shapes (spherical and cylindrical) that characterize small tumor-like inclusions. Numerical simulations show the robustness of the algorithm to reconstruct the anisotropy factor with different contrast levels, inclusion depths, initial guesses, heterogeneous background, noise levels, and two-layered medium. The crosstalk problem when reconstructing simultaneously μ s and g has been reported and achieved with a reasonable quality.

CONCLUSIONS

The proposed RTE-based reconstruction algorithm is robust to spatially retrieve and localize small tumoral inclusions. Heterogeneities in g-factor have been accurately reconstructed which makes the new algorithm a candidate of choice to image this factor as new intrinsic contrast biomarker for optical imaging.

Diffusion Equation-Assisted Markov Chain Monte Carlo Methods for the Inverse Radiative Transfer Equation

Optical tomography is the process of reconstructing the optical properties of biological tissue using measurements of incoming and outgoing light intensity at the tissue boundary. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE), and the inverse problem becomes reconstructing the diffusion coefficient using Dirichlet and Neumann data on the boundary. We study this problem in the Bayesian framework, meaning that we examine the posterior distribution of the scattering coefficient after the measurements have been taken. However, sampling from this distribution is computationally expensive, since to evaluate each Markov Chain Monte Carlo (MCMC) sample, one needs to run the RTE solvers multiple times. We therefore propose the DE-assisted two-level MCMC technique, in which bad samples are filtered out using DE solvers that are significantly cheaper than RTE solvers. This allows us to make sampling from the RTE posterior distribution computationally feasible.

Spatial frequency domain imaging using an analytical model for separation of surface and volume scattering

A method to correct for surface scattering in spatial frequency domain imaging (SFDI) is presented. The use of a modified analytical solution of the radiative transfer equation allows calculation of the reflectance and the phase of a rough semi-infinite geometry so that both spatial frequency domain reflectance and phase can be applied for precise retrieval of the bulk optical properties and the surface scattering. For validation of the method, phantoms with different surface roughness were produced. Contrarily, with the modified theory, it was possible to dramatically reduce systematic errors due to surface scattering. The evaluation of these measurements with the state-of-the-art theory and measuring modality, i.e., using crossed linear polarizers, reveals large errors in the determined optical properties, depending on the surface roughness, of up to ≈100  %  . These results were confirmed with SFDI measurements on a phantom that has a structured rough surface.

Stochastic Proximal Gradient Algorithms for Multi-Source Quantitative Photoacoustic Tomography

The development of accurate and efficient image reconstruction algorithms is a central aspect of quantitative photoacoustic tomography (QPAT). In this paper, we address this issues for multi-source QPAT using the radiative transfer equation (RTE) as accurate model for light transport. The tissue parameters are jointly reconstructed from the acoustical data measured for each of the applied sources. We develop stochastic proximal gradient methods for multi-source QPAT, which are more efficient than standard proximal gradient methods in which a single iterative update has complexity proportional to the number applies sources. Additionally, we introduce a completely new formulation of QPAT as multilinear (MULL) inverse problem which avoids explicitly solving the RTE. The MULL formulation of QPAT is again addressed with stochastic proximal gradient methods. Numerical results for both approaches are presented. Besides the introduction of stochastic proximal gradient algorithms to QPAT, we consider the new MULL formulation of QPAT as main contribution of this paper.

Radiative transfer with delta-Eddington-type phase functions

The radiative transfer equation (RTE) arises in a wide variety of applications, in particular, in biomedical imaging applications associated with the propagation of light through the biological tissue. However, highly forward-peaked scattering feature in a biological medium makes it very challenging to numerically solve the RTE problem accurately. One idea to overcome the difficulty associated with the highly forward-peaked scattering is through the use of a delta-Eddington phase function. This paper is devoted to an RTE framework with a family of delta-Eddington-type phase functions. Significance in biomedical imaging applications of the RTE with delta-Eddington-type phase functions are explained. Mathematical studies of the problems include solution existence, uniqueness, and continuous dependence on the problem data: the inflow boundary value, the source function, the absorption coefficient, and the scattering coefficient. Numerical results are presented to show that employing a delta-Eddington-type phase function with properly chosen parameters provides accurate simulation results for light propagation within highly forward-peaked scattering media.

Radiative transfer equation modeling by streamline diffusion modified continuous Galerkin method

Optical tomography has a wide range of biomedical applications. Accurate prediction of photon transport in media is critical, as it directly affects the accuracy of the reconstructions. The radiative transfer equation (RTE) is the most accurate deterministic forward model, yet it has not been widely employed in practice due to the challenges in robust and efficient numerical implementations in high dimensions. Herein, we propose a method that combines the discrete ordinate method (DOM) with a streamline diffusion modified continuous Galerkin method to numerically solve RTE. Additionally, a phase function normalization technique was employed to dramatically reduce the instability of the DOM with fewer discrete angular points. To illustrate the accuracy and robustness of our method, the computed solutions to RTE were compared with Monte Carlo (MC) simulations when two types of sources (ideal pencil beam and Gaussian beam) and multiple optical properties were tested. Results show that with standard optical properties of human tissue, photon densities obtained using RTE are, on average, around 5% of those predicted by MC simulations in the entire/deeper region. These results suggest that this implementation of the finite element method-RTE is an accurate forward model for optical tomography in human tissues.

X-ray micromodulated luminescence tomography in dual-cone geometry

We propose a scanning method utilizing dual-cone beams of x-rays to induce luminescence from nanophosphors and reconstruct the three-dimensional distribution of these particles in a biological sample or a small animal. For this purpose, x-rays are focused through a polycapillary lens onto a spot of a few micrometers in size. Such x-ray scanning can be point-wise performed to acquire photon emission data on an object surface. The x-ray-induced luminescence data allow for reliable image reconstruction with high spatial resolution and large imaging depth. We describe several numerical simulation studies to demonstrate the feasibility and merits of the proposed approach.

Parallel multigrid solver of radiative transfer equation for photon transport via graphics processing unit

A graphics processing unit-based parallel multigrid solver for a radiative transfer equation with vacuum boundary condition or reflection boundary condition is presented for heterogeneous media with complex geometry based on two-dimensional triangular meshes or three-dimensional tetrahedral meshes. The computational complexity of this parallel solver is linearly proportional to the degrees of freedom in both angular and spatial variables, while the full multigrid method is utilized to minimize the number of iterations. The overall gain of speed is roughly 30 to 300 fold with respect to our prior multigrid solver, which depends on the underlying regime and the parallelization. The numerical validations are presented with the MATLAB codes at https://sites.google.com/site/rtefastsolver/.

Integral equations of the photon fluence rate and flux based on a generalized Delta-Eddington phase function

We present a generalized Delta-Eddington phase function to simplify the radiative transfer equation to integral equations with respect to both photon fluence rate and flux vector. The photon fluence rate and flux can be solved from the system of integral equations. By comparing to the Monte Carlo simulation results, the solutions of the system of integral equations accurately model the photon propagation in biological tissue over a wide range of optical parameters.