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.

Quantitative photoacoustic tomography (QPAT) offers the possibility of highresolution 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 Jacobianbased 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.

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.

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 lowdimensional 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.

With inverse problems there are often several unknown distributed parameters of which only one may be of interest. Since assigning incorrect fixed values to the uninteresting parameters usually leads to a severely erroneous model, one is forced to estimate all distributed parameters simultaneously. This may increase the computational complexity of the problem significantly. In the Bayesian framework, all unknowns are generally treated as random variables and estimated simultaneously and all uncertainties can be modeled systematically. Recently, the approximation error approach has been proposed for handling uncertainty and model-reduction-related errors in the models. In this approach approximate marginalization of these errors is carried out before the estimation of the interesting variables. In this paper we discuss the adaptation of the approximation error approach to the marginalization of uninteresting distributed parameters. As an example, we consider the marginalization of scattering coefficient in diffuse 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.

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.

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