Tanja Tarvainen

Approximation errors and model reduction with an application in optical diffusion tomography

Model reduction is often required in several applications, typically due to limited available time, computer memory or other restrictions. In problems that are related to partial differential equations, this often means that we are bound to use sparse meshes in the model for the forward problem. Conversely, if we are given more and more accurate measurements, we have to employ increasingly accurate forward problem solvers in order to exploit the information in the measurements. Optical diffusion tomography (ODT) is an example in which the typical required accuracy for the forward problem solver leads to computational times that may be unacceptable both in biomedical and industrial end applications. In this paper we review the approximation error theory and investigate the interplay between the mesh density and measurement accuracy in the case of optical diffusion tomography. We show that if the approximation errors are estimated and employed, it is possible to use mesh densities that would be unacceptable with a conventional measurement model.

Coupled radiative transfer equation and diffusion approximation model for photon migration in turbid medium with low-scattering and non-scattering regions.

In this paper, a coupled radiative transfer equation and diffusion approximation model is extended for light propagation in turbid medium with low-scattering and non-scattering regions. The light propagation is modelled with the radiative transfer equation in sub-domains in which the assumptions of the diffusion approximation are not valid. The diffusion approximation is used elsewhere in the domain. The two equations are coupled through their boundary conditions and they are solved simultaneously using the finite element method. The streamline diffusion modification is used to avoid the ray-effect problem in the finite element solution of the radiative transfer equation. The proposed method is tested with simulations. The results of the coupled model are compared with the finite element solutions of the radiative transfer equation and the diffusion approximation and with results of Monte Carlo simulation. The results show that the coupled model can be used to describe photon migration in turbid medium with low-scattering and non-scattering regions more accurately than the conventional diffusion model.

Hybrid radiative-transfer–diffusion model for optical tomography

A hybrid radiative-transfer-diffusion model for optical tomography is proposed. The light propagation is modeled with the radiative-transfer equation in the vicinity of the laser sources, and the diffusion approximation is used elsewhere in the domain. The solution of the radiative-transfer equation is used to construct a Dirichlet boundary condition for the diffusion approximation on a fictitious interface within the object. This boundary condition constitutes an approximative distributed source model for the diffusion approximation in the remaining area. The results from the proposed approach are compared with finite-element solutions of the radiative-transfer equation and the diffusion approximation and Monte Carlo simulation. The results show that the method improves the accuracy of the forward model compared with the conventional diffusion model.

A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation

Quantitative photoacoustic tomography (QPAT) offers the possibility of high-resolution molecular imaging by quantifying molecular concentrations in biological tissue. QPAT comprises two inverse problems: (1) the construction of a photoacoustic image from surface measurements of photoacoustic wave pulses over time, and (2) the determination of the optical properties of the imaged region. The first is a well-studied area for which a number of solution methods are available, while the second is, in general, a nonlinear, ill-posed inverse problem. Model-based inversion techniques to solve (2) are usually based on the diffusion approximation to the radiative transfer equation (RTE) and typically assume the acoustic inversion step has been solved exactly. Here, neither simplification is made: the full RTE is used to model the light propagation, and the acoustic propagation and image reconstruction are included in the simulations of measured data. Since Hessian- and Jacobian-based minimizations are computationally expensive for the large data sets typically encountered in QPAT, gradient-based minimization schemes provide a practical alternative. The acoustic pressure time series were simulated using a k-space, pseudo-spectral time domain model, and a time-reversal reconstruction algorithm was used to form a set of photoacoustic images corresponding to four illumination positions. A regularized, adjoint-assisted gradient inversion using a finite element model of the RTE was then used to determine the optical absorption and scattering coefficients.

Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography

In this article, we describe the multichannel implementation of an intensity modulated optical tomography system developed at Helsinki University of Technology. The system has two time-multiplexed wavelengths, 16 time-multiplexed source fibers and 16 parallel detection channels. The gain of the photomultiplier tubes (PMTs) is individually adjusted during the measurement sequence to increase the dynamic range of the system by 104. The PMT used has a high quantum efficiency in the near infrared (8% at 800nm), a fast settling time, and low hysteresis. The gain of the PMT is set so that the dc anode current is below 80nA, which allows the measurement of phase independently of the intensity. The system allows measurements of amplitude at detected intensities down to 1fW, which is sufficient for transmittance measurements of the female breast, the forearm, and the brain of early pre-term infants. The mean repeatability of phase and the logarithm of amplitude (lnA) at 100MHz were found to be 0.08° and 0.004, res...

An approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography

In this paper, we investigate the applicability of the Bayesian approximation error approach to compensate for the discrepancy of the diffusion approximation in diffuse optical tomography close to the light sources and in weakly scattering subdomains. While the approximation error approach has earlier been shown to be a feasible approach to compensating for discretization errors, uncertain boundary data and geometry, the ability of the approach to recover from using a qualitatively incorrect physical model has not been contested. In the case of weakly scattering subdomains and close to sources, the radiative transfer equation is commonly considered to be the most accurate model for light scattering in turbid media. In this paper, we construct the approximation error statistics based on predictions of the radiative transfer and diffusion models. In addition, we investigate the combined approximation errors due to using the diffusion approximation and using a very low-dimensional approximation in the forward problem. We show that recovery is feasible in the sense that with the approximation error model the reconstructions with a low-dimensional diffusion approximation are essentially as good as with using a very high-dimensional radiative transfer model.

Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography

Quantitative photoacoustic tomography is a novel hybrid imaging technique aiming at estimating optical parameters inside tissues. The method combines (functional) optical information and accurate anatomical information obtained using ultrasound techniques. The optical inverse problem of quantitative photoacoustic tomography is to estimate the optical parameters within tissue when absorbed optical energy density is given. In this paper we consider reconstruction of absorption and scattering distributions in quantitative photoacoustic tomography. The radiative transport equation and diffusion approximation are used as light transport models and solutions in different size domains are investigated. The simulations show that scaling of the data, for example by using logarithmic data, can be expected to significantly improve the convergence of the minimization algorithm. Furthermore, both the radiative transport equation and diffusion approximation can give good estimates for absorption. However, depending on the optical properties and the size of the domain, the diffusion approximation may not produce as good estimates for scattering as the radiative transport equation.

Approximation errors and model reduction in three-dimensional diffuse optical tomography

Model reduction is often required in diffuse optical tomography (DOT), typically because of limited available computation time or computer memory. In practice, this means that one is bound to use coarse mesh and truncated computation domain in the model for the forward problem. We apply the (Bayesian) approximation error model for the compensation of modeling errors caused by domain truncation and a coarse computation mesh in DOT. The approach is tested with a three-dimensional example using experimental data. The results show that when the approximation error model is employed, it is possible to use mesh densities and computation domains that would be unacceptable with a conventional measurement error model.

Computational calibration method for optical tomography.

We propose a computational calibration method for optical tomography. The model of the calibration scheme is based on the rotation symmetry of source and detector positions in the measurement setup. The relative amplitude losses and phase shifts at the optic fibers are modeled by complex-valued coupling coefficients. The coupling coefficients can be estimated when optical tomography data from a homogeneous and isotropic object are given. Once these coupling coefficients have been estimated, any data measured with the same measurement setup can be corrected for the relative variation in the data due to source and detector losses. The final calibration of the data for the source and detector losses and the source calibration between the data and the forward model are obtained as part of the initial estimation for reconstruction. The calibration method was tested with simulations and measurements. The results show that the coupling coefficients of the sources and detectors can be estimated with good accuracy. Furthermore, the results show that the method can significantly improve the quality of reconstructed images.

A Bayesian approach to spectral quantitative photoacoustic tomography

A Bayesian approach to the optical reconstruction problem associated with spectral quantitative photoacoustic tomography is presented. The approach is derived for commonly used spectral tissue models of optical absorption and scattering: the absorption is described as a weighted sum of absorption spectra of known chromophores (spatially dependent chromophore concentrations), while the scattering is described using Mie scattering theory, with the proportionality constant and spectral power law parameter both spatially-dependent. It is validated using two-dimensional test problems composed of three biologically relevant chromophores: fat, oxygenated blood and deoxygenated blood. Using this approach it is possible to estimate the Gr?neisen parameter, the absolute chromophore concentrations, and the Mie scattering parameters associated with spectral photoacoustic tomography problems. In addition, the direct estimation of the spectral parameters is compared to estimates obtained by fitting the spectral parameters to estimates of absorption, scattering and Gr?neisen parameter at the investigated wavelengths. It is shown with numerical examples that the direct estimation results in better accuracy of the estimated parameters.