Ossi Lehtikangas

Finite element approximation of the radiative transport equation in a medium with piece-wise constant refractive index

The radiative transport equation can be used as a light transport model in a medium with scattering particles, such as biological tissues. In the radiative transport equation, the refractive index is assumed to be constant within the medium. However, in biomedical media, changes in the refractive index can occur between different tissue types. In this work, light propagation in a medium with piece-wise constant refractive index is considered. Light propagation in each sub-domain with a constant refractive index is modeled using the radiative transport equation and the equations are coupled using boundary conditions describing Fresnel reflection and refraction phenomena on the interfaces between the sub-domains. The resulting coupled system of radiative transport equations is numerically solved using a finite element method. The approach is tested with simulations. The results show that this coupled system describes light propagation accurately through comparison with the Monte Carlo method. It is also shown that neglecting the internal changes of the refractive index can lead to erroneous boundary measurements of scattered light.

Modeling boundary measurements of scattered light using the corrected diffusion approximation

We study the modeling and simulation of steady-state measurements of light scattered by a turbid medium taken at the boundary. In particular, we implement the recently introduced corrected diffusion approximation in two spatial dimensions to model these boundary measurements. This implementation uses expansions in plane wave solutions to compute boundary conditions and the additive boundary layer correction, and a finite element method to solve the diffusion equation. We show that this corrected diffusion approximation models boundary measurements substantially better than the standard diffusion approximation in comparison to numerical solutions of the radiative transport equation.

Reconstruction of velocity fields in electromagnetic flow tomography

Electromagnetic flow meters (EMFMs) are the gold standard in measuring flow velocity in process industry. The flow meters can measure the mean flow velocity of conductive liquids and slurries. A drawback of this approach is that the velocity field cannot be determined. Asymmetric axial flows, often encountered in multiphase flows, pipe elbows and T-junctions, are problematic and can lead to serious systematic errors. Recently, electromagnetic flow tomography (EMFT) has been proposed for measuring velocity fields using several coils and a set of electrodes attached to the surface of the pipe. In this work, a velocity field reconstruction method for EMFT is proposed. The method uses a previously developed finite-element-based computational forward model for computing boundary voltages and a Bayesian framework for inverse problems. In the approach, the vz-component of the velocity field along the longitudinal axis of the pipe is estimated on the pipe cross section. Different asymmetric velocity fields encountered near pipe elbows, solids-in-water flows in inclined pipes and in stratified or multiphase flows are tested. The results suggest that the proposed reconstruction method could be used to estimate velocity fields in complicated pipe flows in which the conventional EMFMs have limited accuracy. This article is part of the themed issue ‘Supersensing through industrial process tomography’.

Utilizing Fokker-Planck-Eddington approximation in modeling light transport in tissues-like media

Fokker-Planck-Eddington approximation can be used to approximate the radiative transport equation when scattering is forward-peaked. In the approach, forward-peaked scattering probability is approximated using delta functions and smoothly varying Legendre polynomials. In this work, the approximation is used to model light propagation in turbid media with low-scattering regions. The proposed model is tested using simulations, and compared with the radiative transport equation, the diffusion approximation, and the coupled radiative transport - diffusion model. The results show that the Fokker-Planck-Eddington approximation describes light propagation with good accuracy.

Optimal Coil Currents in Electromagnetic Flow Tomography

Electromagnetic flow meters are a gold standard in measuring the mean flow velocity of conductive liquids and slurries in process industry. A drawback of this approach is that the velocity field cannot be determined. Velocity field information is important for characterizing multiphase flows in the process industry. Recently, electromagnetic flow tomography has been proposed for estimating velocity fields in process pipes. The modality uses multiple magnetic field excitations produced by coils and a set of electrodes attached to the inner surface of the pipe to measure the induced voltages. In earlier studies, a method for reconstructing 2-D velocity field on a pipe cross section has been developed. The method utilizes a finite-element-based computational forward model for computing boundary voltages and a Bayesian framework for inverse problem to reconstruct the velocity field. Magnetic field excitations affect the boundary voltage measurements and, hence, the reconstructed velocity field. Optimization of excitations is especially important when imaging axisymmetric flows, since all axisymmetric velocity fields having the same mean velocity produce the same boundary voltage data when uniform magnetic field excitations are used. In this paper, two methods for optimizing coil currents and resulting magnetic fields are proposed. The methods are based on maximizing the norm of the boundary voltage measurements or minimizing the uncertainty in the reconstructed velocity field estimates. The results show that by optimizing coil currents it is possible to obtain accurate velocity field estimates using just one or two optimal excitations.

Utilising the coupled radiative transfer - diffusion model in diffuse optical tomography

The coupled radiative transfer - diffusion model can be used as light transport model in turbid media with non-diffusive regions. In the coupled radiative transfer - diffusion model, light propagation is modelled with the radiative transfer equation in sub-domains in which the approximations of the diffusion equation are not valid and the diffusion approximation is used elsewhere in the domain. In this work, the image reconstruction problem of diffuse optical tomography utilising the coupled radiative transfer - diffusion model is considered. Absorption and scattering distributions are estimated using the coupled radiative transfer - diffusion model as a forward model for light propagation. The results are compared to reconstructions obtained using other light transport models. The results show that the coupled radiative transfer - diffusion model can produce as good estimates for absorption and scattering as the full radiative transfer equation also in situations in which the approximations of the diffusion equation are not valid.