Compensation of modeling errors due to unknown domain boundary in diffuse optical tomography

Effects of atlas-based anatomy on modelled light transport in the neonatal head

Objective.Diffuse optical tomography (DOT) provides a relatively convenient method for imaging haemodynamic changes related to neuronal activity on the cerebral cortex. Due to practical challenges in obtaining anatomical images of neonates, an anatomical framework is often created from an age-appropriate atlas model, which is individualized to the subject based on measurements of the head geometry. This work studies the approximation error arising from using an atlas instead of the neonate's own anatomical model.Approach.We consider numerical simulations of frequency-domain (FD) DOT using two approaches, Monte Carlo simulations and diffusion approximation via finite element method, and observe the variation in (1) the logarithm of amplitude and phase shift measurements, and (2) the corresponding inner head sensitivities (Jacobians), due to varying segmented anatomy. Varying segmentations are sampled by registering 165 atlas models from a neonatal database to the head geometry of one individual selected as the reference model. Prior to the registration, we refine the segmentation of the cerebrospinal fluid (CSF) by separating the CSF into two physiologically plausible layers.Main results.In absolute measurements, a considerable change in the grey matter or extracerebral tissue absorption coefficient was found detectable over the anatomical variation. In difference measurements, a small local 10%-increase in brain absorption was clearly detectable in the simulated measurements over the approximation error in the Jacobians, despite the wide range of brain maturation among the registered models.Significance.Individual-level atlas models could potentially be selected within several weeks in gestational age in DOT difference imaging, if an exactly age-appropriate atlas is not available. The approximation error method could potentially be implemented to improve the accuracy of atlas-based imaging. The presented CSF segmentation algorithm could be useful also in other model-based imaging modalities. The computation of FD Jacobians is now available in the widely-used Monte Carlo eXtreme software.

A Model-Based Iterative Learning Approach for Diffuse Optical Tomography

Diffuse optical tomography (DOT) utilises near-infrared light for imaging spatially distributed optical parameters, typically the absorption and scattering coefficients. The image reconstruction problem of DOT is an ill-posed inverse problem, due to the non-linear light propagation in tissues and limited boundary measurements. The ill-posedness means that the image reconstruction is sensitive to measurement and modelling errors. The Bayesian approach for the inverse problem of DOT offers the possibility of incorporating prior information about the unknowns, rendering the problem less ill-posed. It also allows marginalisation of modelling errors utilising the so-called Bayesian approximation error method. A more recent trend in image reconstruction techniques is the use of deep learning, which has shown promising results in various applications from image processing to tomographic reconstructions. In this work, we study the non-linear DOT inverse problem of estimating the (absolute) absorption and scattering coefficients utilising a 'model-based' learning approach, essentially intertwining learned components with the model equations of DOT. The proposed approach was validated with 2D simulations and 3D experimental data. We demonstrated improved absorption and scattering estimates for targets with a mix of smooth and sharp image features, implying that the proposed approach could learn image features that are difficult to model using standard Gaussian priors. Furthermore, it was shown that the approach can be utilised in compensating for modelling errors due to coarse discretisation enabling computationally efficient solutions. Overall, the approach provided improved computation times compared to a standard Gauss-Newton iteration.

Modeling of Errors Due to Uncertainties in Ultrasound Sensor Locations in Photoacoustic Tomography

Photoacoustic tomography is an imaging modality based on the photoacoustic effect caused by the absorption of an externally introduced light pulse. In the inverse problem of photoacoustic tomography, the initial pressure generated through the photoacoustic effect is estimated from a measured photoacoustic time-series utilizing a forward model for ultrasound propagation. Due to the ill-posedness of the inverse problem, errors in the forward model or measurements can result in significant errors in the solution of the inverse problem. In this work, we study modeling of errors caused by uncertainties in ultrasound sensor locations in photoacoustic tomography using a Bayesian framework. The approach is evaluated with simulated and experimental data. The results indicate that the inverse problem of photoacoustic tomography is sensitive even to small uncertainties in sensor locations. Furthermore, these uncertainties can lead to significant errors in the estimates and reduction of the quality of the photoacoustic images. In this work, we show that the errors due to uncertainties in ultrasound sensor locations can be modeled and compensated using Bayesian approximation error modeling.

Time-domain diffuse optical tomography utilizing truncated Fourier series approximation

Diffuse optical tomography (DOT) uses near infrared light for in vivo imaging of spatially varying optical parameters in biological tissues. It is known that time-resolved measurements provide the richest information on soft tissues, among other measurement types in DOT such as steady-state and intensity-modulated measurements. Therefore, several integral-transform-based moments of the time-resolved DOT measurements have been considered to estimate spatially distributed optical parameters. However, the use of such moments can result in low-contrast images and cross-talks between the reconstructed optical parameters, limiting their accuracy. In this work, we propose to utilize a truncated Fourier series approximation in time-resolved DOT. Using this approximation, we obtained optical parameter estimates with accuracy comparable to using whole time-resolved data that uses low computational time and resources. The truncated Fourier series approximation based estimates also displayed good contrast and minimal parameter cross-talk, and the estimates further improved in accuracy when multiple Fourier frequencies were used.

Modelling of errors due to speed of sound variations in photoacoustic tomography using a Bayesian framework

Inverse problem of estimating initial pressure in photoacoustic tomography is ill-posed and thus sensitive to errors in modelling and measurements. In practical experiments, accurate knowledge of the speed of sound of the imaged target is commonly not available, and therefore an approximate speed of sound is used in the computational model. This can result in errors in the solution of the inverse problem that can appear as artefacts in the reconstructed images. In this paper, the inverse problem of photoacoustic tomography is approached in a Bayesian framework. Errors due to uncertainties in the speed of sound are modelled using Bayesian approximation error modelling. Estimation of the initial pressure distribution together with information on the reliability of these estimates are considered. The approach was studied using numerical simulations. The results show that uncertainties in the speed of sound can cause significant errors in the solution of the inverse problem. However, modelling of these uncertainties improves the accuracy of the solution.

Image Reconstruction with Reliability Assessment in Quantitative Photoacoustic Tomography

Quantitative photoacoustic tomography is a novel imaging method which aims to reconstruct optical parameters of an imaged target based on initial pressure distribution, which can be obtained from ultrasound measurements. In this paper, a method for reconstructing the optical parameters in a Bayesian framework is presented. In addition, evaluating the credibility of the estimates is studied. Furthermore, a Bayesian approximation error method is utilized to compensate the modeling errors caused by coarse discretization of the forward model. The reconstruction method and the reliability of the credibility estimates are investigated with two-dimensional numerical simulations. The results suggest that the Bayesian approach can be used to obtain accurate estimates of the optical parameters and the credibility estimates of these parameters. Furthermore, the Bayesian approximation error method can be used to compensate for the modeling errors caused by a coarse discretization, which can be used to reduce the computational costs of the reconstruction procedure. In addition, taking the modeling errors into account can increase the reliability of the credibility estimates.

Three dimensional photoacoustic tomography in Bayesian framework

The image reconstruction problem (or inverse problem) in photoacoustic tomography is to resolve the initial pressure distribution from detected ultrasound waves generated within an object due to an illumination by a short light pulse. Recently, a Bayesian approach to photoacoustic image reconstruction with uncertainty quantification was proposed and studied with two dimensional numerical simulations. In this paper, the approach is extended to three spatial dimensions and, in addition to numerical simulations, experimental data are considered. The solution of the inverse problem is obtained by computing point estimates, i.e., maximum a posteriori estimate and posterior covariance. These are computed iteratively in a matrix-free form using a biconjugate gradient stabilized method utilizing the adjoint of the acoustic forward operator. The results show that the Bayesian approach can produce accurate estimates of the initial pressure distribution in realistic measurement geometries and that the reliability of these estimates can be assessed.

Contrast enhancement in EIT imaging of the brain

We consider electrical impedance tomography (EIT) imaging of the brain. The brain is surrounded by the poorly conducting skull which has low conductivity compared to the brain. The skull layer causes a partial shielding effect which leads to weak sensitivity for the imaging of the brain tissue. In this paper we propose an approach based on the Bayesian approximation error approach, to enhance the contrast in brain imaging. With this approach, both the (uninteresting) geometry and the conductivity of the skull are embedded in the approximation error statistics, which leads to a computationally efficient algorithm that is able to detect features such as internal haemorrhage with significantly increased sensitivity and specificity. We evaluate the approach with simulations and phantom data.

Nonlinear approach to difference imaging in diffuse optical tomography

Difference imaging aims at recovery of the change in the optical properties of a body based on measurements before and after the change. Conventionally, the image reconstruction is based on using difference of the measurements and a linear approximation of the observation model. One of the main benefits of the linearized difference reconstruction is that the approach has a good tolerance to modeling errors, which cancel out partially in the subtraction of the measurements. However, a drawback of the approach is that the difference images are usually only qualitative in nature and their spatial resolution can be weak because they rely on the global linearization of the nonlinear observation model. To overcome the limitations of the linear approach, we investigate a nonlinear approach for difference imaging where the images of the optical parameters before and after the change are reconstructed simultaneously based on the two datasets. We tested the feasibility of the method with simulations and experimental data from a phantom and studied how the approach tolerates modeling errors like domain truncation, optode coupling errors, and domain shape errors.